Aymeric Blot
email: aymeric.blot@univ-lille.fr

Manuel López-Ibáñez
email: manuel.lopez-ibanez@manchester.ac.uk

Marie-Éléonore Kessaci
email: mkessaci@univ-lille.fr

Laetitia Jourdan
email: laetitia.jourdan@univ-lille.fr

Archive-aware Scalarisation-based Multi-Objective Local Search For a Bi-objective Permutation Flowshop Problem

Keywords: Flowshop scheduling, Local search, Heuristics, Multiobjective optimisation, Combinatorial optimisation

Given the availability of high-performing local search (LS) for single-objective (SO) optimisation problems, one successful approach to tackle their multi-objective (MO) counterparts is scalarisation-based local search (SBLS). SBLS strategies solve multiple scalarisations, i.e., aggregations of the multiple objectives into a single scalar value, with varying weights. They have been shown to work specially well as the initialisation phase of other types of multi-objective local search, such as Pareto local search (PLS). A major drawback of existing SBLS strategies is that the underlying SO optimiser is unaware of the MO nature of the problem and only returns a single solution, discarding any intermediate solutions that may be of interest. We propose here two new SBLS strategies (ChangeRestart and Change-Direction) that overcome this drawback by augmenting the underlying SO-LS method with an archive of nondominated solutions that is used to dynamically update the scalarisations. The new strategies produce better results on the bi-objective permutation flowshop problem than other five SBLS strategies from the literature, not only on their own but also when used as the initialisation phase of PLS.

Introduction

Multi-objective (MO) local search methods [START_REF] Paquete | Stochastic local search algorithms for multiobjective combinatorial optimization: A review[END_REF][START_REF] Liefooghe | On dominancebased multiobjective local search: design, implementation and experimental analysis on scheduling and traveling salesman problems[END_REF][START_REF] Dubois-Lacoste | Combining two search paradigms for multi-objective optimization: Two-Phase and Pareto local search[END_REF] are usually classified into two types. Scalarisation-based local search (SBLS) strategies aggregate the multiple objectives into a single (scalar) one by means of weights, and use singleobjective (SO) local search to tackle each scalarised problem. Dominance-based local search (DBLS) strategies search the neighbourhood of candidate solutions for (Pareto) dominating or nondominated solutions. Successful algorithms for MO combinatorial optimisation problems often hybridise both strategies by generating a set of high-quality solutions by means of SBLS, and further improving this set by applying a DBLS method [START_REF] Dubois-Lacoste | A hybrid TP+PLS algorithm for bi-objective flow-shop scheduling problems[END_REF][START_REF] Liefooghe | On dominancebased multiobjective local search: design, implementation and experimental analysis on scheduling and traveling salesman problems[END_REF][START_REF] Lust | The multiobjective multidimensional knapsack problem: a survey and a new approach[END_REF][START_REF] Dubois-Lacoste | Combining two search paradigms for multi-objective optimization: Two-Phase and Pareto local search[END_REF].

Various SBLS strategies have been proposed in the literature that mainly differ in the sequence of weights explored during the search and the starting solution for solving each scalarisation. The simplest method, henceforth called Restart [START_REF] Paquete | A two-phase local search for the biobjective traveling salesman problem[END_REF], uses a uniform set of weights and starts each scalarisation from a randomly (or heuristically) generated solution. More advanced strategies, such as AdaptiveAnytime [START_REF] Dubois-Lacoste | Improving the anytime behavior of two-phase local search[END_REF], dynamically compute the next weight and choose a starting solution among the best ones found so far with the goal of closing the largest "gap" in the current Pareto front approximation.

First, we propose to augment the single-objective local search that solves the scalarisations with an archive of nondominated solutions, such that they are able to return more than one solution and make these solutions visible to the overall SBLS strategy and to the other local search runs solving other scalarisations. Then, we propose two new SBLS strategies. With ChangeRestart, we subdivide the time granted to solve each scalarisation in multiple steps, and use intermediary solutions to restart each local search run when it falls behind. With ChangeDirection, we further improve ChangeRestart by changing not only the starting solution of a local search run, but also the weight that defines the scalarisation being solved. As a case study, we focus on a bi-objective variant of the permutation flowshop scheduling problem (PFSP), which has been used previously as a benchmark for MO local search [START_REF] Dubois-Lacoste | A hybrid TP+PLS algorithm for bi-objective flow-shop scheduling problems[END_REF].

This paper is organised as follows. Section 2 describes classical SBLS strategies, together with the PFSP variant on which they have been previously applied. Section 3 proposes to augment single-objective local search with a nondominated archive to be used within SBLS strategies; and Section 4 proposes two new SBLS strategies. The experimental setup is then detailed in Section 5, whose results are discussed in Section 6. Section 7 summarises the main conclusions.

Background

Scalarisation-based Local Search (SBLS)

In multi-objective combinatorial optimisation problems, we have a set of feasible solutions S, where each solution may be evaluated according to a vector of M objectives f (s) = (f 1 (s), . . . , f M (s)). Without a priori information, candidate solutions are usually compared in terms of Pareto dominance:

s 1 dominates s 2 iff ∀i = 1, . . . , M , f i (s 1 ) ≤ f i (s 2 ) and ∃j, f j (s 1 ) < f j (s 2
). The goal becomes to find, or approximate as well as possible, the Pareto-optimal set, i.e., the set of solutions S * ⊂ S that are not dominated by any other solution in S. The image of the Pareto-optimal set in the objective space is called the Pareto front.

A MO problem can be transformed into a SO one by scalarising it, for example, by means of weighted sum. For simplicity, we will focus on the biobjective (M = 2) case in the following. Given a problem with two objectives f (s) = (f 1 (s), f 2 (s)) and a normalised weight vector λ = (λ, 1 -λ), where λ ∈ [0, 1] ⊂ R, the corresponding scalarised problem (scalarisation) is computed as

f λ (s) = λ • f 1 (s) + (1 -λ) • f 2 (s).
An optimal solution of this SO scalarisation is a Pareto-optimal solution of the MO problem, thus multiple Pareto-optimal solutions (although maybe not all) may be obtained by solving multiple scalarisations with different weights. The main advantage of solving scalarisations instead of the original MO problem is that, very often, highly effective and efficient local search algorithms exist for the single-objective case. SBLS approaches are conceptually related to decomposition-based algorithms (e.g., MOEA/D [START_REF] Zhang | MOEA/D: A multiobjective evolutionary algorithm based on decomposition[END_REF]).

Literature SBLS strategies differ in how weights are generated and which solution is used as the starting point of each local search run LS λ solving f λ .

Perhaps the simplest restarting strategy consists in generating a set of uniformly distributed weights and start each LS λ run from a randomly or heuristically generated solution.

TPLS. In the simplest version of TPLS [START_REF] Paquete | A two-phase local search for the biobjective traveling salesman problem[END_REF], one high-quality solution is generated by optimising just the first objective. On a second phase, a sequence of scalarisations of the problem, with weights that increasingly favours the second objective, are tackled by running LS λ , thus generating solutions along the Pareto frontier from the first to the second objective. Moreover, each run of LS λ starts from the best solution found for the previous scalarisation. This strategy is called 1to2 or 2to1 depending on which objective is optimised in the first phase. The 1to2 and 2to1 strategies tend to produce better solutions towards the starting objective. In order to avoid this bias, an alternative strategy (Double) uses half of the weights for 1to2 and the other half for 2to1 [START_REF] Paquete | A two-phase local search for the biobjective traveling salesman problem[END_REF][START_REF] Dubois-Lacoste | Improving the anytime behavior of two-phase local search[END_REF] AdaptiveAnytime. The above TPLS strategies can lead to uneven exploration of the objective space and poorly distributed approximation of the Pareto front if the problem is not fairly regular in terms of difficulty and the Pareto front is roughly symmetric for all scalarising directions. Similar poor results will also be obtained if the algorithm is terminated before tackling the predefined number of scalarisations. The adaptive anytime TPLS (AdaptiveAnytime) strategy was proposed to address these issues [START_REF] Dubois-Lacoste | Improving the anytime behavior of two-phase local search[END_REF]. Similar to TPLS, a first phase generates one high-quality solution for each individual objective (like in Double) and a second phase solves a sequence of scalarisations. AdaptiveAnytime maintains a set G of "gaps" in the current approximation to the Pareto front, where each gap is a pair of solutions that are neighbours in the objective space, i.e., no other solution exists within the hyper-cube defined by them, and the size of the gap is the volume of this hyper-cube. The most successful variant of AdaptiveAnytime solves two scalarisations at each step, by first finding the largest gap in G, e.g., (s 1 , s 2 ), with f 1 (s 1 ) < f 1 (s 2 ), then computing:

λ 1 = λ -θ • λ λ 2 = λ + θ • (1 -λ) where λ = f 2 (s 1 ) -f 2 (s 2 ) f 2 (s 1 ) -f 2 (s 2 ) + f 1 (s 2 ) -f 1 (s 1 ) (1)
and θ ∈ [0, 1] is a parameter that biases λ 1 towards the first objective and λ 2 towards the second objective; and, finally, solving f λ1 starting from s 1 and f λ2 starting from s 2 . The solution returned by solving each scalarisation is used to update G, by removing any dominated solutions and updating the corresponding gaps. Thus, each step of the AdaptiveAnytime strategy tries to reduce the size of the largest gap and adapt the weights to the shape of the current front.

Bi-objective Permutation Flowshop Scheduling

The above SBLS strategies have been tested on various bi-objective permutation flowshop scheduling problems (PFSP) [START_REF] Dubois-Lacoste | Improving the anytime behavior of two-phase local search[END_REF] and AdaptiveAnytime was later used as the initialisation phase of the state-of-the-art MO local search [START_REF] Dubois-Lacoste | A hybrid TP+PLS algorithm for bi-objective flow-shop scheduling problems[END_REF].

The PFSP is among the best-known problems in the scheduling literature, since it models several typical problems in manufacturing. Given a set of n jobs to be processed sequentially on m machines, where each job requires a different processing time on each machine, the goal is to find a permutation of the jobs that optimises particular objectives, such that all the jobs are processed in the same order on all machines, and the order of the machines is the same for all jobs. In this paper, we focus on the bi-objective variant (bPFSP) that minimises the makespan, that is, the completion time of the last job, and the total flowtime, that is, the sum of completion times of all jobs.

Archive-aware SBLS Strategies

Traditional SBLS strategies, such as the ones described above (Restart, 1to2, 2to1, Double and AdaptiveAnytime), use a SO local search to find a new solution optimised for a given scalarisation. Each local search run (LS λ ) starts from a given solution and returns the single best solution found for that particular scalarisation. Any other solution found during the run is usually discarded, even those solutions that are not dominated by the solution returned.

We propose to augment the SO local search with an archive that keeps track of nondominated solutions found while solving a scalarisation, in order to preserve solutions that may be optimal for the MO problem, even if they are not for the particular scalarisation. Since such intermediary solutions are fully evaluated to compute their scalarised value, keeping an archive of these solutions only adds the computational overhead of updating the archive. In practice, adding every solution evaluated to the archive would require too much time. Instead, we only update the archive when a new solution replaces the current one.

As an example of SO local search, let us consider iterated greedy (IG) [START_REF] Ruiz | A simple and effective iterated greedy algorithm for the permutation flowshop scheduling problem[END_REF]. At each iteration of IG, the current solution π is randomly destructed (e.g, by removing some jobs from it), heuristically reconstructed (e.g., by re-inserting the jobs in new positions), and the resulting solution may be further improved by another local search. An acceptance criterion replaces the current solution (π) with the new one if the latter is better or some other condition is met. In any case, if the new solution improves the best-so-far one (π ), the latter is replaced. The algorithm returns π once it terminates.

Our proposed archive-aware IG adds an archive of nondominated solutions (A) that is updated every time a better current solution is found, and returns the archive in addition to the best solution found. Any other SO local search used within SBLS strategies can be made archive-aware in a similar manner.

We propose straight-forward variants of the classical SBLS strategies that make use of such archive-aware SO local search and we denote such variants with the suffix " arch ". In Restart arch , 1to2 arch , 2to1 arch and Double arch , each local search run produces an archive instead of a single solution. The resulting N scalar archives are independent of each other until merged into a final archive. Thus, the search trajectory of these archive-aware SBLS variants is the same as their original counterparts, except for the overhead incurred by updating the archives. In the case of AdaptiveAnytime arch , the archive returned by each local search run is immediately merged with the overall archive so that all solutions returned by the local search are used for computing the next largest gap.

New SBLS Strategies: ChangeRestart, ChangeDirection

In this section, we propose two new SBLS strategies that make use of intermediate solutions found while solving the scalarised problems, and that may also use an archive-aware single-objective local search.

ChangeRestart

We observed that the sub-spaces searched by running the SO local search for different values of λ often overlap, thus the best-so-far solution found for one scalarisation may be worse than the best-so-far solution found for another, when the latter solution is evaluated on the former scalarisation. Thus, the main idea behind ChangeRestart is to divide each local search run (LS λ ) into smaller steps and, at each step, decide to either continue the run until the next step or restart it from a new solution. When interrupted, LS λ returns its best-so-far solution (π λ ). Then, for each weight, we calculate the scalarised value of all solutions in the current nondominated archive A. If the best-so-far solution by a given scalarised run is actually worse than the best-so-far solution found by a different scalarised run, when the latter solution is evaluated on the scalarised problem being solved by the former run, then this former run restarts its search from the latter solution. By reducing the number of steps (N steps ), we can limit the computational overhead of recalculating the scalarised values of each solution in A for all weights. In particular, the time limit assigned to each LS λ run is divided by N steps . When N steps = 1, ChangeRestart is identical to Restart.

In the archive-aware variant ChangeRestart arch , each run of LS λ returns a nondominated archive that is merged with the overall archive A.

Figure 1 shows possible executions of ChangeRestart and ChangeRestart arch for two scalarisations and three steps (N steps = 3). Blue points ( ) and red triangles ( ) show the initial solutions and the best solutions found after each step. These solutions are connected with arrows to show the trajectory followed by each run of LS λ . Unfilled points ( ) and triangles ( ) show intermediary solutions in the archive after each step. For ChangeRestart (left), after the second step, the solution (a) found for λ = 1 has a worse value in the first objective than the solution (b) found for λ = 0.5. Thus, the local search for λ = 1 re-starts from solution b instead of a. For ChangeRestart arch (right), the local search re-starts instead from solution (c), as it has an even better value regarding objective f 1 .

ChangeDirection

The second SBLS strategy proposed here is ChangeDirection. While ChangeRestart is an extension of Restart, ChangeDirection is inspired by the more advanced AdaptiveAnytime. In AdaptiveAnytime, scalarisation weights are dynamically chosen according to the gaps in the current overall archive (A), in order to focus the search in the direction which will most improve the current approximation to the Pareto front. In ChangeDirection, as in ChangeRestart, the runs of LS λ are also divided in a number of steps and, after each step, solutions from different scalarisations are merged into an overall archive A. However, instead of only updating the starting solution of each LS λ run, the scalarisation weight λ is also updated.

In other words, in addition to speeding up an LS λ run by re-starting from a better initial solution, the scalarisation direction of LS λ may be changed to focus on the largest gap in the current approximation front. In particular, a scalarisation weight λ is replaced by another weight whenever the best-so-far of LS λ is worse, according to f λ , than a solution returned by another local search run. In that case, the computational resources allocated to searching in the direction given by λ could be better used in searching on a different direction.

ChangeDirection only differs from ChangeRestart in the deletion and replacement of scalarisation directions. Thus, we will only explain those novel parts. First, we delete those scalarisation weights for which the best solution found in the last run of LS λ is worse, according to the same scalarisation f λ , than a solution in A. Then, following the strategy of AdaptiveAnytime explained earlier, the gaps in the current approximation front are computed and new weights are generated from the largest gap to replace the deleted ones. In particular, two weights are generated from each gap (Eq. 1) until all deleted weights are replaced. When only one additional weight is needed, it is chosen randomly between the two weights produced by the gap. The new scalarisations then start from the solutions constituting the sides of the gap. If all gaps are used and additional weights are needed, they are drawn uniformly at random within [0, 1] and initial solutions are taken uniformly at random from A. Finally, as in Change-Restart, each LS λ either re-starts from a new initial solution if its scalarisation was introduced in this step, or continues from its current solution, otherwise. As previously, in the archive-aware variant ChangeDirection arch , each run of LS λ returns a nondominated archive that is merged with the overall archive A.

Experimental Setup

We wish to investigate not only whether the new proposed SBLS strategies work well on their own, but also if they provide a good initial set for a dominancebased local search (DBLS) algorithm. Thus, we use the various SBLS strategies as the initialisation phase of a hybrid of SBLS+DBLS algorithm, where a SBLS strategy generates an initial approximation front that is further improved by a DBLS strategy, in our case, an iterated variant of Pareto local search (IPLS). We use IG as the single-objective local search (LS λ ) and the algorithms are evaluated on the bi-objective PFSP (bPFSP). In this section, we explain the details of the our experimental setup.

bPFSP Instances. As a benchmark, we consider the well-known Taillard instances [START_REF] Taillard | Benchmarks for basic scheduling problems[END_REF]. In the following, we focus on 80 instances with 20, 50, 100 and 200 jobs, and 10 and 20 machines, 10 instances being used for each valid combination.

Iterated Greedy (IG). The single-objective local search used by the SBLS strategies is Iterative greedy (IG) [START_REF] Ruiz | A simple and effective iterated greedy algorithm for the permutation flowshop scheduling problem[END_REF], which is a state-of-the-art algorithm for the single-objective PFSP. The particular IG variant and parameter settings are directly taken from the bPFSP literature [START_REF] Dubois-Lacoste | A hybrid TP+PLS algorithm for bi-objective flow-shop scheduling problems[END_REF]. For the archive-aware SBLS strategies, we augment this IG variant with an archive as explained in Section 3.

Iterated Pareto Local Search (IPLS).

As the DBLS component of our hybrid SBLS+DBLS algorithm, we consider an iterated variant of Pareto local search (PLS) [START_REF] Paquete | Pareto local optimum sets in the biobjective traveling salesman problem: An experimental study[END_REF], as it was shown that even simple perturbations could benefit PLS algorithms [START_REF] Drugan | Path-guided mutation for stochastic Pareto local search algorithms[END_REF]. Our iterated PLS (IPLS) extends the PLS used in [START_REF] Dubois-Lacoste | A hybrid TP+PLS algorithm for bi-objective flow-shop scheduling problems[END_REF] by perturbing the archive when the latter converges to a Pareto local optimal set, using the generalised framework of [START_REF] Blot | Automatic design of multi-objective local search algorithms: case study on a bi-objective permutation flowshop scheduling problem[END_REF]. The perturbation used creates a new archive by taking every current solution and replacing it with one of its neighbours, taken uniformly at random, three times in a row; dominated solutions from this new set are then filtered. As the neighbourhood of PLS, we use the union of the exchange and insertion neighbourhoods [START_REF] Dubois-Lacoste | Anytime Pareto local search[END_REF], in which two positions of two jobs are swapped and one job is reinserted at another position, respectively.

Termination criteria. The termination criterion of algorithms applied to the bPFSP is usually set as maximum running time proportional to the instance size. While previous works have used a linear proportion of both the number of jobs n and the number of machines m (e.g., 0.1 • n • m CPU seconds [START_REF] Dubois-Lacoste | A hybrid TP+PLS algorithm for bi-objective flow-shop scheduling problems[END_REF]), we use a maximum running time for the hybrid SBLS+IPLS of 0.002 • n 2 • m CPU seconds. Indeed, the total number of solutions grows exponentially and the typical size of permutation neighbourhoods grows quadratically, making a linear running time less relevant. The coefficient 0.002 was chosen to match the linear formula for n = 50 and m = 20. As for the termination criterion PLS itself, we use the number of successive iterations without improvement, set here to n, before forcing a perturbation of the archive.

The SBLS strategies are limited to 25% of the total running time of the hybrid SBLS+IPLS, and the remaining 75% is allocated to IPLS. The main parameter of the SBLS strategies is the number of scalarisations (N scalar ), that is, the number of runs of IG executed in addition to two individual runs for each of the two single objectives. Following [START_REF] Dubois-Lacoste | A hybrid TP+PLS algorithm for bi-objective flow-shop scheduling problems[END_REF], we perform longer runs of IG for the two single objectives (IG {1,2} ) than for the other scalarisations (IG λ ), with the time assigned to IG {1,2} being 1.5 times the time assigned to IG λ . As more time is allocated to IG {1,2} than to IG Λ , their respective running time budgets are 1.5/(N scalar +3) and 1/(N scalar +3) of the total time assigned to the SBLS strategy. In the case of ChangeRestart and ChangeDirection, the maximum runtime of each IG is further divided by N steps .

The following experiments are separated in three successive phases. First, we analyse the effect of using an archive-aware IG on the five SBLS strategies from the literature (Restart, 1to2, 2to1, Double, and AdaptiveAnytime). Second, we compare all these SBLS variants with the new SBLS strategies proposed here (ChangeRestart and ChangeDirection), including their archive-aware counterparts. Finally, we analyse other possible setting for the parameters N scalar and N steps . Unless stated otherwise, ChangeRestart and ChangeDirection use N steps = 20; all SBLS strategies use a fixed value of N scalar = 12; and both AdaptiveAnytime and ChangeDirection use θ = 0.25 for Eq. 1 [START_REF] Dubois-Lacoste | A hybrid TP+PLS algorithm for bi-objective flow-shop scheduling problems[END_REF].

In all cases, we run the hybrid SBLS+IPLS and we save the archive returned by the SBLS strategies (before IPLS) and the final archive (after IPLS). Each experiment is repeated 5 times, using independent random seeds, on each of the 80 Taillard instances. All replications use the same seeds on the same instances. All the experiments have been conducted on Intel Xeon E5-2687W V4 CPUs (3.0GHz, 30MB cache, 64GB RAM).

Results are evaluated according to both the hypervolume and the additiveε indicators [START_REF] Zitzler | Performance assessment of multiobjective optimizers: an analysis and review[END_REF]. Indicator values have been computed independently on every instance aggregating all results generated for the instance and scaling both objectives to a 0-1 scale in which 0 (1) corresponds to the minimum (maximum) objective value reached by any solution. The hypervolume variant 1-HV is used, with 0 corresponding to the maximum hypervolume, so that both indicators are to be minimised. The reference point used for computing the hypervolume indicator is (1.0001, 1.0001). The reference set for computing additive-ε indicator is the set of nondominated solutions from all aggregated results for each instance.

Experimental Results

Known SBLS strategies vs. their archive-aware variants

First, we compare the five SBLS strategies from the literature with their archiveaware variants. Figure 2 shows the mean hypervolume and additive-ε values obtained by each strategy before and after running IPLS. Averages are computed on the indicator values over all 80 bPFSP instances.

For both indicators, all the proposed archive-aware variants (in red) lead to improved quality before IPLS. After results are improved by IPLS, all archiveaware variants produce again better results than their original counter-parts, with the exception of AdaptiveAnytime. This is somewhat surprising and further analysis is needed to understand this behaviour. Interestingly, some of the archive-aware variants are able to outperform AdaptiveAnytime when their original variants are not.

Performance of Two New SBLS strategies

We now compare the newly proposed SBLS strategies (ChangeRestart and Change-Direction) to the ones from the literature as well as their archive-aware variants.

In terms of hypervolume, all four new strategies achieve in average much better results than the strategies from the literature, with ChangeRestart arch achieving the best results both on its own and when further improved by IPLS. However, in terms of additive-ε, the non-archive-aware ChangeRestart strategy performs much worse than the three other new strategies, but still better than most strategies from the literature. Overall, the best strategy according to both indicators appears to be the archive-aware ChangeRestart strategy.

To validate these observations, Table 1 shows the results of a statistical analysis comparing all approaches, without averaging over benchmarks, for both hypervolume (top) and additive ε (bottom). For each benchmark set, and for all possible pairs of strategies, we conducted a statistical Wilcoxon test comparing their final quality (after IPLS) paired on 50 runs. A () in the table indicates strategies for which there was no other strategy performing statistically better (with 95% confidence). In other words, within each row, all strategies with are not statistically better to each other, while for those strategies without a , there was at least one other strategy statistically better. As shown in the Table, the SBLS strategies from the literature are often outperformed by some other strategy, whereas their archive-aware variants are less often so, in particular on the smaller instances with 20 and 50 jobs. Finally, ChangeRestart arch and both variants of ChangeDirection are almost never outperformed, even on the largest instances, validating our previous observations. SBLS strategies strongly depends on the number of scalarisations they use. Our choice of N scalar = 12 was motivated by previous studies that few [START_REF] Paquete | A two-phase local search for the biobjective traveling salesman problem[END_REF]20) scalarisations should be preferred [START_REF] Dubois-Lacoste | A hybrid TP+PLS algorithm for bi-objective flow-shop scheduling problems[END_REF]. Figure 3 (left) shows for the 14 previous strategies the final performance regarding both hypervolume and additive ε indicators, for both parameter values of N scalar ∈ {6, 12} scalarisations, in order to see the impact of archives-aware mechanisms when using very few scalarisations.

We can see that for all strategies, both with and without archiving, using 12 scalarisations improve in average significantly the performance regarding hypervolume and slightly the one regarding the ε indicator, hinting that even with archiving a sufficient number of scalarisations are still required.

The number of steps, i.e. how many times we can restart the scalarisations, is at the core of the two new SBLS strategies we propose. Figure 3 As the figure shows, at first increasing the number of steps from one largely improves the quality of the results. Increasing the number of steps further specially benefits the archive-aware variants and in particular, ChangeDirection arch . However, for large values of N steps , the quality improvements stop or, in several cases, worsen. Thus, it appears that even larger values would not improve the results reported here.

Conclusion

This paper proposes and evaluates two complementary ways of augmenting scalarisation-based local search (SBLS) strategies by making the underlying single-objective local search aware of the multi-objective nature of the problem. Our first proposal adds an archive of nondominated solutions to the singleobjective local search. Our results showed that these archive-aware SBLS variants always improve over their original counterparts when ran on their own. Moreover, this improvement also shows for nearly all SBLS strategies when their results are further improved by means of iterated Pareto local search (IPLS).

Our second proposal was to divide each run of the single-objective local search into a number of smaller steps and, at each step, restart scalarisations that produce poor results. We proposed two SBLS strategies that differ on what is changed by the restart. In ChangeRestart, the local search for solving a scalarisation is restarted from the best-known solution for that scalarisation problem. This solution was possibly generated when solving a different scalarisation. In ChangeDirection, not only the starting solution, but also the weight that defines the scalarisation problem itself being solved are both updated in order to re-focus this particular run on the largest gap of the current approximation front.

Our experimental results show that these two new SBLS strategies outperform five classical SBLS strategies from the literature, even when the latter are using and archive-aware local search. In particular, ChangeDirection produces consistently the best results, either on its own or when used as the initialisation phase of a hybrid SBLS+IPLS algorithm, which suggests that the new strategies may lead to new state-of-the-art results for the bi-objective permutation flowshop [START_REF] Dubois-Lacoste | A hybrid TP+PLS algorithm for bi-objective flow-shop scheduling problems[END_REF], and other problems. An additional benefit of ChangeDirection is that it maintains the adaptive behaviour of AdaptiveAnytime, while it also may perform N scalar local search runs in parallel between steps. Future work will analyse in more detail the interaction between the new SBLS strategies and the archive-aware SO local search. A more comprehensive analysis of the effect of the N scalar and N steps parameters would be needed to understand their interactions with problem features. We would also hope to evaluate the new proposals in terms of their anytime behaviour [START_REF] Dubois-Lacoste | Improving the anytime behavior of two-phase local search[END_REF]. Finally, a further comparison, including various common speedups not included here as we focused on archive-aware mechanisms, would be required for a fair comparison with other state-of-the-art algorithms.

5 Fig. 1 .

 51 Fig. 1. Example runs of ChangeRestart (left) and ChangeRestart arch (right) (N steps = 3).

Fig. 2 .

 2 Fig. 2. Comparison of all SBLS strategies according to (left) mean hypervolume and (right) mean additive-ε.

  (right) shows for all variants of the ChangeRestart and ChangeDirection strategies the impact of the parameter N steps , for values of N steps = 1 (equivalent to Restart) and N steps ∈ {2, 5, 10, 15, 20, 25}, using the final performance regarding both the hypervolume and ε indicators. Marks indicate the value of N steps , while colours indicate the strategy.

Table 1 .

 1 SBLS strategies not statistically outperformed by another strategy after IPLS step, using paired Wilcoxon tests (left: hypervolume; right: additive-ε) Analysis of Parameters N scalar and N steps

	Restart	1to2	2to1	Double	AdaptiveAnytime	Restart arch	1to2 arch	2to1 arch	Double arch	AdaptiveAnytime arch	ChangeRestart	ChangeRestart arch	ChangeDirection	ChangeDirection arch	Restart	1to2	2to1	Double	AdaptiveAnytime	Restart arch	1to2 arch	2to1 arch	Double arch	AdaptiveAnytime arch	ChangeRestart	ChangeRestart arch	ChangeDirection	ChangeDirection arch
	20 × 10
	20 × 20
	50 × 10
	50 × 20
	100 × 10
	100 × 20
	200 × 10
	200 × 20
	6.3

  Impact of the number of scalarisations (left) and the number of steps (right)

		0.11
						0.105
	ε after IPLS	0.1 0.105 0.095			ε after IPLS	0.095 0.1	1 2 5 10 15 20 25
						0.09
		0.16	0.17	0.18		0.16	0.17
		1 -HV after IPLS			1 -HV after IPLS
		N scalar = 6 N scalar = 6 (arch)	ChangeRestart ChangeDirection ChangeDirection arch ChangeRestart arch
		N scalar = 12 N scalar = 12 (arch)		Restart
	Fig. 3.


L Sylla
email: sylla.lamine@ugb.edu.sn

Keywords: Integro-partial differential equation, Backward stochastic differential equations with jumps, Viscosity solution, Non-local operator, Concavity. MSC 2010 subject classifications: 35D40, 35R09, 60H30

Another method of viscosity solutions of integro-differential partial equation by concavity

Lamine Sylla

To cite this version:

Lamine Sylla. Another method of viscosity solutions of integro-differential partial equation by concavity. 2018.

Introduction

We consider the following system of integro-partial differential equation, which is a function of (t, x): ∀i ∈ {1, . . . , m},        -∂ t u i (t, x)b(t, x) ⊤ D x u i (t, x) -1 2 Tr(σσ ⊤ (t, x)D 2 xx u i (t, x)) -K i u i (t, x) -h (i) (t, x, u i (t, x), (σ ⊤ D x u i )(t, x), B i u i (t, x)) = 0, (t, x) ∈ [0, T ] × R k ; u i (T, x) = g i (x);

(1. [START_REF] Barles | Backward stochastic differential equations and integralpartial differential equations[END_REF] where the operators B i and K i are defined as follows:

B i u i (t, x) = E γ i (t,
x, e)(u i (t, x + β(t, x, e))u i (t, x))λ(de);

(1.2)

K i u i (t, x) = E (u i (t, x + β(t,
x, e))u i (t, x)β(t, x, e) ⊤ D x u i (t, x))λ(de).

The resolution of (1.1) is in connection with the following system of backward stochastic differential equations with jumps :

   dY i;t,x s = -f (i) (s, X t,
Y i;t,x T = g i (X t,x T ); (1.3)
and the following standard stochastic differential equation of diffusion-jump type:

X t,x s = x + s t b(r, X t,x r ) dr + s t σ(r, X t,x r ) dB r + s t E β(r, X t,x r-, e) µ(dr, de), (1.4) for s ∈ [t, T ] and X t,x s = x if s ≤ t.

Several authors have studied (1.1) by various and varied methods. Among others we can quote Barles and al. [START_REF] Barles | Backward stochastic differential equations and integralpartial differential equations[END_REF], who use the theorem of comparison by supposition the monotony of the generator; Ouknine and Hamadène [START_REF] Hamadène | Reflected backward stochastic differential equation with jumps and random obstacle[END_REF] use the penalization method.

But recently Hamadène does a relaxation on the hypothesis of monotony on the generator to introduce a new class of functions(see [START_REF] Hamadène | Viscosity solutions of second order integral-partial differential equations without monotonicity condition: A new result[END_REF] page 216) for the resolution of (1.1).

In this work we propose to solve (1.1) by relaxing the monotonicity of the generator and the class of fonctions introduced in [START_REF] Hamadène | Viscosity solutions of second order integral-partial differential equations without monotonicity condition: A new result[END_REF] and assuming that λ = ∞ and the concavity of the generator at the level of its second variable. We recall that this technic was used in [START_REF]L p solutions of multidimensional BSDEs with weak monotonicity and general growth generators[END_REF] for the resolution of BSDE.

Our paper is organized as follows: in the next section we give the notations and the assumptions; in section 3 we recall a number of existing results; in section 4 we build estimates and properties for a good resolution of our problem; section 5 is reserved to give our main result.

And at the end, classical definition of the concept of viscosity solution is put in appendix.

Notations and assumptions

Let (Ω, F , (F t ) t≤T , P) be a stochastic basis such that F 0 contains all P-null sets of F , and F t = F t+ := ǫ>0 F t+ǫ , t ≥ 0, and we suppose that the filtration is generated by the two mutually independents processes:

(i) B := (B t ) t≥0 a d-dimensional Brownian motion and, (ii) a Poisson random measure µ on R + × E where E := R ℓ -{0} is equipped with its Borel field E (ℓ ≥ 1).

The compensator ν(dt, de) = dtλ(de) is such that { µ([0, t] × A) = (µλ)([0, t] × A)} t≥0 is a martingale for all A ∈ E satisfying λ(A) < ∞. We also assume that λ is a σ-finite measure on (E, E), integrates the function (1∧ | e | 2 ) and λ(E) = ∞.

Let's now introduce the following spaces:

(iii) P (resp. P) the field on [0, T ] × Ω of F t≤T -progressively measurable (resp. predictable) sets.

(iv) For κ ≥ 1, L 2 κ (λ) the space of Borel measurable functions ϕ := (ϕ(e)

) e∈E from E into R κ such that ϕ 2 L 2 κ (λ) = E |ϕ(e)| 2 κ λ(de) < ∞; L 2
1 (λ) will be simply denoted by L 2 (λ); (v) S 2 (R κ ) the space of RCLL (for right continuous with left limits) P-measurable and R κ -valued pro-

cesses such that E[sup s≤T |Y s | 2 ] < ∞;
(vi) H 2 (R κ×d ) the space of processes Z := (Z s ) s≤T which are P-measurable, R κ×d -valued and satisfying

E T 0 |Z s | 2 ds < ∞; (vii) H 2 (L 2 κ (λ)
) the space of processes U := (U s ) s≤T which are P-measurable, L 2 κ (λ)-valued and satisfying

E T 0 U s (ω) 2 L 2 κ (λ) ds < ∞; (viii) Π g the set of deterministics functions ̟ : (t, x) ∈ [0, T ] × R κ → ̟(t, x) ∈ R of
polynomial growth, i.e., for which there exists two non-negative constants C and p such that for any

(t, x) ∈ [0, T ] × R κ , |̟(t, x)| ≤ C(1 + |x| p ).
The subspace of Π g of continuous functions will be denoted by Π c g ; (ix) M the class of functions which satisfy the p-order Mao condition in x i.e. If f ∈ M then there exists a nondecreasing, continuous and concave function ρ(•) : R + → R + with ρ(0) = 0, ρ(u) > 0, for u > 0 and

0 + du ρ(u) = +∞, such that dP × dt-a.e., ∀x, x ′ ∈ R k and ∀p ≥ 2, |f (t, x, y, z, q) -f (t, x ′ , y, z, q)| ≤ ρ 1 p (|x -x ′ | p );
(x) For any process θ := (θ s ) s≤T and t ∈ (0, T ], θ t-= lim sրt θ s and

∆ t θ = θ t -θ t-.
Now let b and σ be the following functions:

b : (t, x) ∈ [0, T ] × R k → b(t, x) ∈ R k ; σ : (t, x) ∈ [0, T ] × R k → σ(t, x) ∈ R k×d .
We assume that b and σ belong to M (2.5)

Let β : (t, x, e) ∈ [0, T ] × R k × E → β(t,
x, e) ∈ R k be a measurable function such that for some real constant C, and for all e ∈ E,

(i) |β(t, x, e)| ≤ C(1 ∧ |e|); (2.6) (ii) β belongs to M; (iii) the mapping (t, x) ∈ [0, T ] × R k → β(t, x, e) ∈ R k is continuous for any e ∈ E.
The functions (g i ) i=1,m and (h (i) ) i=1,m be two functions defined as follows: for i = 1, . . . , m,

g i : R k -→ R m x -→ g i (x)
and

h (i) : [0, T ] × R k+m+d+1 -→ R (t, x, y, z, q) -→ h (i) (t, x, y, z, q).
Moreover we assume they satisfy:

(H1): For any i ∈ {1, . . . , m}, the function g i belongs to M.

(H2): For any i ∈ {1, . . . , m}, (i) the function h (i) is Lipschitz in (y, z, q) uniformly in (t, x), i.e., there exists a real constant C such that for any (t, x) ∈ [0, T ] × R k , (y, z, q) and (y ′ , z ′ , q ′ ) elements of R m+d+1 ,

h (i) (t, x, y, z, q) -h (i) (t, x, y ′ , z ′ , q ′ ) ≤ C(|y -y ′ | + |z -z ′ | + |q -q ′ |); (2.7)
(ii) the (t, x) → h (i) (t, x, y, z, q), for fixed (y, z, q) ∈ R m+d+1 , belongs uniformly to M, Next let γ i , i = 1, . . . , m be Borel measurable functions defined from [0, T ]×R k ×E into R and satisfying:

(i) γ i (t, x, e) ≤ C(1 ∧ |e|); (ii) γ i belongs to M; (2.8) (iii) the mapping t ∈ [0, T ] → γ i (t, x, e) ∈ R is continuous for any (x, e) ∈ R k × E.
Finally we introduce the following functions (f (i) ) i=1,m defined by:

∀(t, x, y, z, ζ) ∈ [0, T ]×R k+m+d ×L 2 (λ), f (i) (t, x, y, z, ζ) := h (i) t, x, y, z, E γ i (t,
x, e)ζ(e)λ(de) . (2.9)

The functions (f (i) ) i=1,m , enjoy the two following properties:

(a) The function f (i) is Lipschitz in (y, z, ζ) uniformly in (t, x), i.e., there exists a real constant C such that

f (i) (t, x, y, z, ζ) -f (i) (t, x, y ′ , z ′ , ζ ′ ) ≤ C(|y -y ′ | + |z -z ′ | + ζ -ζ ′ L 2 (λ) );
(2.10) since h (i) is uniformly Lipschitz in (y, z, q) and γ i verifies (2.8)-(i);

(b) The function (t, x) ∈ [0, T ] × R k → f (i) (t, x, 0, 0, 0) belongs to Π c g ;
and then

E T 0 f (i) (r, X t,x r , 0, 0, 0) 2 dr < ∞.
3 Some results in backward stochastic differential equation with jumps

A class of diffusion processes with jumps

Let (t, x) ∈ [0, T ] × R d and (X t,x s ) s≤T be the stochastic process solution of (1.4). Under assumptions (2.5)-(2.6) the solution of Equation (1.4) exists and is unique (see [START_REF] Fujiwara | Stochastic differential equations of jump type and Lévy processes in differomorphism group[END_REF] for more details). We state some properties of the process {(X t,x s ), s ∈ [0, T ]} which can found in [START_REF] Fujiwara | Stochastic differential equations of jump type and Lévy processes in differomorphism group[END_REF].

Proposition 3.1 For each t ≥ 0, there exists a version of {(X t,x s ), s ∈ [t, T ]} such that s → X t s is a C 2 (R d )-valued rcll process. Moreover it satisfies the following estimates: ∀p ≥ 2, x, x ′ ∈ R d and s ≥ t, E[ sup t≤r≤s X t,x r -x p ] ≤ M p (s -t)(1 + |x| p ); E[ sup t≤r≤s X t,x r -X t,x ′ r -(x -x ′ ) p p ] ≤ M p (s -t)(|x -x ′ | p ); (3.11)
for some constant M p .

Existence and uniqueness for BSDE with jumps

Let (t, x) ∈ [0, T ] × R d and we consider the following m-dimensional BSDE with jumps: For proof of this proposition we can see [START_REF] Barles | Backward stochastic differential equations and integralpartial differential equations[END_REF].

         (i) Y t,x := (Y i,t,x ) i=1,m ∈ S 2 (R m ), Z t,x := (Z i,t,x ) i=1,m ∈ H 2 (R m×d ), U t,x := (U i,t,x ) i=1,m ∈ H 2 (L 2 m (λ)); (ii) dY i;t,x s = -f (i) (s, X t,

Viscosity solutions of integro-differential partial equation

Proposition 3.3 (see, [START_REF] Barles | Backward stochastic differential equations and integralpartial differential equations[END_REF]) Assume that (H1), (H2), are fulfilled. Then there exists deterministic continuous functions (u i (t, x)) i=1,m which belong to Π g such that for any (ii) for any fixed (t, x, y, z) ∈ [0, T ] × R k+m+d , the mapping

(t, x) ∈ [0, T ] × R k ,
(q ∈ R) -→ h (i) (t, x, y, z, q) ∈ R is non-decreasing.
The function (u i ) i=1,m is a continuous viscosity solution (in Barles and al. 's sense, see Definition 5.3 in the Appendix) of (1.1). The solution (u i ) i=1,m of (1.1) is unique in the class Π c g .

Remark 1 (see, [START_REF] Barles | Backward stochastic differential equations and integralpartial differential equations[END_REF]) Under the assumptions (H1), (H2), there exists a unique viscosity solution of (1.1) in the class of functions satisfying

lim |x|→+∞ |u(t, x)| e -A[log(|x|)] 2 = 0 (3.14)
uniformly for t ∈ [0, T ], for some A > 0.

Estimates and properties

In this section, we will establish a priori estimates concerning solutions of BSDE (3.12), which will play an important role in the proof of our main results.

Lemma 4.1 (see, [START_REF] Hamadène | Viscosity solutions of second order integral-partial differential equations without monotonicity condition: A new result[END_REF]) Under assumption (H1), (H2), for any p ≥ 2 there exists two non-negative constants C and δ such that,

E   T 0 ds E U t,x s (e) 2 λ(de) p 2   = E   T 0 ds U t,x s 2 L 2 m (λ) p 2   ≤ C 1 + |x| δ . (4.15) Proposition 4.2 For any i = 1, . . . , m, there exist C ≥ 0, κ > 0 such that, ∀ x and x ′ elements of R k |u i (t, x) -u i (t, x ′ )| 2 ≤ ρ M 2 |x -x ′ | 2 (1 + |x -x ′ | 2 ) [C(1 + |x| κ )] Proof. Let x and x ′ be elements of R k . Let ( Y t,x , Z t,x , U t,x ) (resp. ( Y t,x ′ , Z t,x ′ , U t,x ′
)) be the solution of the BSDE with jumps (3.12) associated with (f (s,

X t,x s , y, η, ζ), g(X t,x T )) (resp. f (s, X t,x ′ s , y, η, ζ), g(X t,x ′ T )). Applying Itô formula to Y t,x -Y t,x ′ 2
between s and T , we have

Y t,x s -Y t,x ′ s 2 + T s |∆Z r | 2 dr + s≤r≤T (∆ r Y t,x r ) 2 (4.16) = g(X t,x T ) -g(X t,x ′ T ) 2 + 2 T s < Y t,x s -Y t,x ′ s , ∆f (r) > dr -2 T s E Y t,x r -Y t,x ′ r (∆U r (e)) µ(dr, de) -2 T s Y t,x r -Y t,x ′ r (∆Z r ) dB r ;
and taking expectation we obtain: ∀s ∈ [t, T ],

E Y t,x s -Y t,x ′ s 2 + T s |∆Z r | 2 dr + T s ∆U r 2 L 2 (λ) dr (4.17) ≤ E g(X t,x T ) -g(X t,x ′ T ) 2 + 2 T s < Y t,x s -Y t,x ′ s , ∆f (r) > dr
where the processes ∆X r , ∆Y r , ∆f (r), ∆Z r and ∆U r are defined as follows:

∀r ∈ [t, T ], ∆f (r) := ((∆f (i) (r)) i=1,m = (f (i) (r, X i;t,x r , Y t,x r , Z i;t,x r , U i;t,x r )-f (i) (r, X i;t,x ′ r , Y i;t,x ′ r , Z i;t,x ′ r , U i;t,x ′ r )) i=1,m , ∆X r = X t,x r -X t,x ′ r , ∆Y (r) = Y t,x r -Y t,x ′ r = (Y j;t,x r -Y j;t,x ′ r ) j=1,m , ∆Z r = Z t,x r -Z t,x ′ r and ∆U r = U t,x r -U t,x ′ r (< •, • > is the usual scalar product on R m
). Now we will give an estimation of each three terms of the second member of inequality (4.17).

• As for any i ∈ {1, . . . , m} g i belongs to M ( for p = 2); therefore

E g(X t,x T ) -g(X t,x ′ T ) 2 ≤ E ρ X t,x T -X t,x ′ T 2 ≤ ρ E X t,x T -X t,x ′ T 2
(by Jensen's inequality)

≤ ρ E X t,x T -X t,x ′ T -(x -x ′ ) 2 + (x -x ′ ) 2 2
and by subsequently using the triangle inequality, the relation (3.11) of proposition 3.1 and the fact that

(a + b) p ≤ 2 p-1 (a p + b p ). E g(X t,x T ) -g(X t,x ′ T ) 2 ≤ ρ M 2 |x -x ′ | 2 (1 + |x -x ′ | 2 ) , (4.18)
• To complete our estimation of (4.17) we need to deal with E 2

T s < Y t,x s -Y t,x ′ s , ∆f (r) > dr .
Taking into account the expression of f (i) given by (2.9) we then split ∆f (r) in the follows way: for

r ≤ T , ∆f (r) = (∆f (r)) i=1,m = ∆ 1 (r) + ∆ 2 (r) + ∆ 3 (r) + ∆ 4 (r) = (∆ i 1 (r) + ∆ i 2 (r) + ∆ i 3 (r) + ∆ i 4 (r)) i=1,m ,
where for any i = 1, . . . , m,

∆ i 1 (r) = h (i) r, X t,x r , Y t,x r , Z i;t,x r , E γ i (r, X t,x r , e)U i;t,x r (e)λ(de) -h (i) r, X t,x ′ r , Y t,x r , Z i;t,x r , E γ i (r, X t,x r , e)U i;t,x r (e)λ(de) ; ∆ i 2 (r) = h (i) r, X t,x ′ r , Y t,x r , Z i;t,x r , E γ i (r, X t,x r , e)U i;t,x r (e)λ(de) -h (i) r, X t,x ′ r , Y t,x ′ r , Z i;t,x r , E γ i (r, X t,x
r , e)U i;t,x r (e)λ(de) ;

∆ i 3 (r) = h (i) r, X t,x ′ r , Y t,x ′ r , Z i;t,x r , E γ i (r, X t,x r , e)U i;t,x r (e)λ(de) -h (i) r, X t,x ′ r , Y t,x ′ r , Z i;t,x ′ r , E γ i (r, X t,x r , e)U i;t,x
r (e)λ(de) ;

∆ i 4 (r) = h (i) r, X t,x ′ r , Y t,x ′ r , Z i;t,x ′ r , E γ i (r, X t,x r , e)U i;t,x r (e)λ(de) -h (i) r, X t,x ′ r , Y t,x ′ r , Z i;t,x ′ r , E γ i (r, X t,x ′ r , e)U i;t,x ′ r (e)λ(de) .
By Cauchy-Schwartz inequality, the inequality 2ab ≤ ǫa 2 + 1 ǫ b 2 , (H2)-(ii), the estimate (3.11) and Jensen's inequality we have:

E 2 T s <∆Y (r),∆1(r)> dr ≤ E 1 ǫ T s |∆Y (r)| 2 dr+ǫ T s ρ |X t,x r -X t,x ′ r | 2 dr ≤ E 1 ǫ T s |∆Y (r)| 2 dr + ǫρ M 2 |x -x ′ | 2 (1 + |x -x ′ | 2 ) . (4.19)
Besides since h (i) is Lipschitz w.r.t. (y, z, q) then,

E 2 T s < ∆Y (r), ∆ 2 (r) > dr ≤ 2CE T s |∆Y (r)| 2 dr , (4.20)
and

E 2 T s < ∆Y (r), ∆ 3 (r) > dr ≤ E 1 ǫ T s |∆Y (r)| 2 dr + C 2 ǫ T s |∆Z(r)| 2 dr . (4.21)
It remains to obtain a control of the last term. But for any s ∈ [t, T ] we have,

E 2 T s < ∆Y (r), ∆ 4 (r) > dr (4.22) ≤ 2CE T s |∆Y (r)| dr × E γ(r, X t,x r , e)U t,x r (e) -γ(r, X t,x ′ r , e)U t,x ′ r (e) λ(de) .
Next by splitting the crossing terms as follows γ(r, X t,x r , e)U t,x r (e)-γ(r,

X t,x ′ r , e)U t,x ′ r (e) = ∆U s (e)γ(s, X t,x s , e)+ U t,x ′ s γ(s, X t,x
s , e)γ(s, X t,x ′ s , e) and setting ∆γ s (e) := γ(s, X t,x s , e)γ(s, X t,x ′ s , e) , we obtain,

E 2 T s < ∆Y (r), ∆ 4 (r) > dr ≤ 2CE T s |∆Y (r)|×   E (|U
2 dr ≤ E T s dr E |U t,x ′ r (e)| 2 λ(de) E |∆γ r (e)| 2 λ(de) ≤ ρ M 2 |x -x ′ | 2 (1 + |x -x ′ | 2 ) × E T s dr E |U t,x ′ r (e)| 2 λ(de) ≤ Cρ M 2 |x -x ′ | 2 (1 + |x -x ′ | 2 ) (1 + |x| κ ). ( 4
E Y t,x s -Y t,x ′ s 2 + T s |∆Z r | 2 dr + T s ∆U r 2 L 2 (λ) dr ≤ E g(X t,x T ) -g(X t,x ′ T ) 2 + 2 T s < Y t,x s -Y t,x ′ s , ∆f (r) > dr ≤ ρ M 2 |x -x ′ | 2 (1 + |x -x ′ | 2 ) C(1 + |x| κ ) + 1 + ǫ + ǫC 3 + 4 ǫ + 2C E T s |∆Y (r)| 2 dr +C 2 ǫE T s |∆Z(r)| 2 dr + C 3 ǫE T s dr E (|∆U r (e)| 2 λ(de) .
By choosing ǫ such that {ǫ + 4 ǫ + ǫ(2C 3 + C 2 ) + 2C < 1} we deduce the existence of a constant C ≥ 0 such that for any s ∈ [t, T ],

E |∆Y (s)| 2 ≤ ρ M 2 |x -x ′ | 2 (1 + |x -x ′ | 2 ) [C(1 + |x| κ )] + E T s |∆Y (r)| 2 dr
and by Gronwall lemma this implies that for any s ∈ [t, T ],

E |∆Y (s)| 2 ≤ ρ M 2 |x -x ′ | 2 (1 + |x -x ′ | 2 ) [C(1 + |x| κ )] .
Finally in taking s = t and considering (3.13) we obtain the desired result. Now we start a point which giving difference of definition viscosity solution between [START_REF] Barles | Backward stochastic differential equations and integralpartial differential equations[END_REF] and [START_REF] Hamadène | Viscosity solutions of second order integral-partial differential equations without monotonicity condition: A new result[END_REF].

It should also be noted that in this part will appear our first contribution after of course the first corresponding to the proposition 4.2. It will be mainly about the use of the M class and the Bihari inequality as in [START_REF]L p solutions of multidimensional BSDEs with weak monotonicity and general growth generators[END_REF].

Proposition 4.3 For any i = 1, . . . , m, (t, x) ∈ [0, T ] × R k , U i;t,x s (e) = u i (s, X t,x s-+ β(s, X t,x s-, e)) -u i (s, X t,x s-), dP ⊗ ds ⊗ dλ -a.e. on Ω × [t, T ] × E. ( 4
µ k (ds, de) = 1 {|e|≥ 1 k } µ(ds, de) and ν k (ds, de) = 1 {|e|≥ 1 k } ν(ds, de).
Which means that, as usual, µ k (ds, de) := (µ k -ν k )(ds, de), is the associated random martingale measure.

The main point to notice is that

λ k (E) = E λ k (de) = E 1 {|e|≥ 1 k } λ(de) = {|e|≥ 1 k } λ(de) = λ({|e| ≥ 1 k }) < ∞. (4.27)
As in [START_REF] Hamadène | Viscosity solutions of second order integral-partial differential equations without monotonicity condition: A new result[END_REF], let us introduce the process k X t,x solving the following standard SDE of jump-diffusion type:

k X t,x s = x + s t b(r, k X t,x r ) dr + s t σ(r, k X t,x r ) dB r + s t E β(r, k X t,x r-, e) µ k (dr, de), t ≤ s ≤ T ; k X t,x r = x if s ≤ t. (4.28)
Note that thanks to the assumptions on b, σ, β the process k X t,x exists and is unique. Moreover it satisfies the same estimates as in (3.11) since λ k is just a truncation at the origin of λ which integrates

(1 ∧ |e| 2 ) e∈E .
On the other hand let us consider the following Markovian BSDE with jumps

                                 (i) E sup s≤T k Y t,x s 2 + T s k Z t,x r 2 dr + T s k U t,x r 2 L 2 (λ k ) dr < ∞ (ii) k Y t,x := ( k Y i,t,x ) i=1,m ∈ S 2 (R m ), k Z t,x := ( k Z i,t,x ) i=1,m ∈ H 2 (R m×d ), k U t,x := ( k U i,t,x ) i=1,m ∈ H 2 (L 2 m (λ k )); (iii) k Y t,x s = g( k X t,x T ) + T s f µ k (r, k X t,x r , k Y t,x r , k Z t,x r , k U t,x r ) dr - T s k Z t,x r dB r + E { k U t,x r (e)} µ k (dr, de) , s ≤ T ; (iv) k Y t,x T = g( k X t,x T ). (4.29)
Finally let us introduce the following functions (f (i) ) i=1,m defined by: ∀

(t, x, y, z, ζ) ∈ [0, T ] × R k × R m × R m×d × L 2 m (λ k ), f µ k (t, x, y, z, ζ) = (f (i) µ k (t, x, y, z i , ζ i )) i=1,m := h (i) t, x, y, z, E γ i (t, x, e)ζ i (e)λ k (de) i=1,m .
First let us emphasize that this latter BSDE is related to the filtration (F k s ) s≤T generated by the Brownian motion and the independent random measure µ k . However this point does not raise major issues since for any s ≤ T , F k s ⊂ F s and thanks to the relationship between µ and µ k . Next by the properties of the functions b, σ, β and by the same opinions of proposition 3.2 and proposition 3.3, there exists an unique quadriple ( k Y t,x , k Z t,x , k U t,x ) solving (4.29) and there also exists a function

u k from [0, T ] × R k into R m of Π c g such that ∀s ∈ [t, T ], k Y t,x := u k (s, k X t,x ), P -a.s. (4.30)
Moreover as in proposition 4.2, there exists positive constants C and κ wich do not depend on k such that:

∀t, x, x ′ , |u k (t, x) -u k (t, x ′ )| ≤ ρ M 2 |x -x ′ | 2 (1 + |x -x ′ | 2 ) [C(1 + |x| κ )] . (4.31)
Finally as λ k is finite then we have the following relationship between the process k U t,x := ( k U i;t,x ) i=1,m and the deterministics functions u k := (u k i ) i=1,m (see [START_REF] Hamadène | Viscosity solutions for second order integro-differential equations without monotonicity condition: The probabilistic Approach[END_REF]): ∀i = 1, . . . , m;

k U i;t,x s (e) = u k i (s, k X t,x s-+ β(s, k X t,x s-, e)) -u k i (s, k X t,x s-), dP ⊗ ds ⊗ dλ k -a.e. on Ω × [t, T ] × E.
This is mainly due to the fact that k U t,x belongs to

L 1 ∩ L 2 (ds ⊗ dP ⊗ dλ k ) since λ k (E) < ∞ and then
we can split the stochastic integral w.r.t. µ k in (4.29). Therefore for all i = 1, . . . , m,

k U i;t,x s (e)1 {|e|≥ 1 k } = (u k i (s, k X t,x s-+β(s, k X t,x s-, e))-u k i (s, k X t,x s-))1 {|e|≥ 1 k } , dP⊗ds⊗dλ k -a.e. on Ω×[t, T ]×E. (4.32)
Step 2: Convergence of the auxiliary processes Let's now prove the following convergence result;

E sup s≤T Y t,x s -k Y t,x s 2 + T 0 Z t,x s -k Z t,x s 2 ds + T 0 ds E λ(de) U t,x s (e) -k U t,x s (e)1 {|e|≥ 1 k } 2 -→ k-→+∞ 0; (4.33)
where (Y t,x , Z t,x , U t,x ) is solution of the BSDE with jumps (3.12).

It should be noted that this convergence (4.33) requires as the technique used in (4.2) the following convergence:

E sup s≤T X t,x s -k X t,x s 2 -→ k-→+∞ 0. (4.34) Proof. [of 4.34] X t,x s -k X t,x s = s 0 (b(r, X t,x r ) -b(r, k X t,x r )) dr + s 0 (σ(r, X t,x r ) -σ(r, k X t,x r )) dB r + s 0 E (β(r, X t,x r-, e) -β(r, k X t,x r-, e)1 {|e|≥ 1 k } ) µ k (dr, de).
E sup 0≤s≤η X t,x s -k X t,x s 2 ≤ 3E sup 0≤s≤η s 0 (b(r, X t,x r ) -b(r, k X t,x r )) dr 2 + sup 0≤s≤η s 0 (σ(r, X t,x r ) -σ(r, k X t,x r )) dB r 2 + sup 0≤r≤η r 0 E (β(r, X t,x r-, e) -β(r, k X t,x r-, e)1 {|e|≥ 1 k } ) µ k (dr, de) 2 ≤ CE η 0 sup 0≤η≤r (b(r, X t,x r ) -b(r, k X t,x r )) 2 + (σ(r, X t,x r ) -σ(r, k X t,x r )) 2 dr +CE η 0 E sup 0≤r≤η β(r, X t,x r-, e) -β(r, k X t,x r-, e) 2 λ k (de)dr + η 0 E sup 0≤r≤η β(r, X t,x r-, e) 2 1 {|e|< 1 k } λ(de)dr (4.36)
Since b, σ and β belong to M, then we have:

∀r ∈ [0, T ], sup 0≤τ ≤r (b(r, X t,x τ ) -b(r, k X t,x τ )) 2 + (σ(r, X t,x τ ) -σ(r, k X t,x τ )) 2 ≤ Cρ( X t,x τ -k X t,x τ 2 ) (4.37)
and

E sup 0≤r≤η β(r, X t,x r-, e) -β(r, k X t,x r-, e) 2 λ k (de) ≤ Cρ( X t,x τ -k X t,x τ 2 ) (4.38)
Plug now those two last inequalities in the previous one to obtain: ∀η ∈ [0, T ],

E sup 0≤s≤η X t,x s -k X t,x s 2 ≤ CE η 0 ρ( X t,x τ -k X t,x τ 2 ) dr + {|e|< 1 k } (1 ∧ |e| 2 ) λ(de) ≤ C η 0 ρ(E X t,x τ -k X t,x τ 2 ) dr + CE {|e|< 1 k }
(1 ∧ |e| 2 ) λ(de) (by Jensen), By Bihari's inequality (see [START_REF]L p solutions of multidimensional BSDEs with weak monotonicity and general growth generators[END_REF] page 171 and [START_REF] Pachpatte | On generalizations of Bihari's inequality[END_REF]) and the fact of

{|e|< 1 k }
(1 ∧ |e| 2 ) λ(de) -→ k-→+∞ 0; we obtain our result the (4.34).

We now focus on (4.33). Note that we can apply Ito's formula, even if the BSDEs are related to filtrations and Poisson random measures which are not the same, since:

(i) F k s ⊂ F s , ∀s ≤ T ; (ii) for any s ≤ T , s 0 k E U i;t,x (e) µ k (dr, de) = s 0 k E U i;t,x (e)1 {|e|≥ 1
k } µ (dr, de) and then the first

(F k s ) s≤T -martingale is also an (F s ) s≤T -martingale. ∀s ∈ [0, T ], Y t,x s -k Y t,x s 2 + T 0 Z t,x s -k Z t,x s 2 ds + s≤r≤T ( k ∆ r Y t,x r ) 2 = g(X t,x T ) -g( k X t,x T ) 2 + 2 T s Y t,x r -k Y t,x r × k ∆f (r) dr -2 T s E Y t,x r -k Y t,x r k ∆U r (e) µ(dr, de) -2 T s Y t,x r -k Y t,x r k ∆Z r dB r ;
and taking expectation we obtain: ∀s ∈ [t, T ],

E Y t,x s -k Y t,x s 2 + T 0 Z t,x s -k Z t,x s 2 + E U t,x s -k U t,x s 1 {|e|≥ 1 k } 2 λ(de) ds ≤ E g(X t,x T ) -g( k X t,x T ) 2 + 2 T s Y t,x r -k Y t,x r × k ∆f (r) dr ; (4.39)
where the processes k ∆X r , k ∆Y r , k ∆f (r), k ∆Z r and k ∆U r are defined as follows: ∀r ∈ [0, T ],

k ∆f (r) := (( k ∆f (i) (r)) i=1,m = (f (i) (r, X t,x r , Y t,x r , Z i;t,x r , U i;t,x r ) -f (i) k (r, k X t,x r , k Y t,x r , k Z t,x r , k U t,x r )) i=1,m , k ∆X r = X t,x r -k X t,x r , k ∆Y (r) = Y t,x r -k Y t,x r = (Y j;t,x r -k Y j;t,x r ) j=1,m , k ∆Z r = Z t,x r -k Z t,x r and k ∆U r = U t,x r -k U t,x s 1 {|e|≥ 1 k } .
Next let us set for r ≤ T ,

k ∆f (r) = (f (r, X t,x r , Y t,x r , Z t,x r , U t,x r ) -f k (r, k X t,x r , k Y t,x r , k Z t,x r , k U t,x r )) = A(r) + B(r) + C(r) + D(r);
where for any i = 1, . . . , m,

A(r) = h (i) r, X t,x r , Y t,x r , Z i;t,x r , E γ i (r, X t,x r , e)U i;t,x r (e)λ(de) -h (i) r, k X t,x r , Y t,x r , Z i;t,x r , E γ i (r, X t,x r , e)U i;t,x r (e)λ(de) i=1,m ; B(r) = h (i) r, k X t,x r , Y t,x r , Z i;t,x r , E γ i (r, X t,x r , e)U i;t,x r (e)λ(de) -h (i) r, k X t,x r , k Y t,x r , Z i;t,x r , E γ i (r, X t,x r , e)U i;t,x r (e)λ(de) i=1,m ; C(r) = h (i) r, k X t,x r , k Y t,x r , Z i;t,x r , E γ i (r, X t,x r , e)U i;t,x r (e)λ(de) -h (i) r, k X t,x r , k Y t,x r , k Z i;t,x r , E γ i (r, X t,x r , e)U i;t,x r (e)λ(de) i=1,m ; D(r) = h (i) r, k X t,x r , k Y t,x r , k Z i;t,x r , E γ i (r, X t,x r , e)U i;t,x r (e)λ(de) -h (i) r, k X t,x r , k Y t,x r , k Z i;t,x r , E γ i (r, k X t,x r , e) k U i;t,x r (e)λ k (de) i=1,m
.

Since g belongs to M and by (4.34) we have,

E g(X t,x T ) -g( k X t,x T ) 2 -→ 0 k→+∞ (4.40) Now we will interest to E T s Y t,x r -k Y t,x r
× k ∆f (r) dr for found (4.33).

By (2.7), we have:

∀r ∈ [0, T ] |A(r)| 2 ≤ ρ( X t,x r -k X t,x r 2 )
;

|B(r)| + |C(r)| ≤ C{ Y t,x r -k Y t,x r + Z t,x r -k Z t,x r }. (4.41)
Now let us deal with D(r) which is more involved. First note that D(r) = (D i (r)) i=1,m where

D i (r) = h (i) r, k X t,x r , k Y t,x r , k Z i;t,x r , E γ i (r, X t,x r , e)U i;t,x r (e)λ(de) -h (i) r, k X t,x r , k Y t,x r , k Z i;t,x r , E γ i (r, k X t,x r , e) k U i;t,x r (e)λ k (de) .
But as h (i) is Lipschitz w.r.t to the last component q and by the relation (2.8) then ,

|D(r)| 2 ≤ C E γ i (r, X t,x r , e)U i;t,x r (e) -γ i (r, k X t,x r , e) k U i;t,x r (e)1 {|e|≥ 1 k } λ(de) 2 ≤ C E γ i (r, X t,x r , e) -γ i (r, k X t,x r , e) U i;t,x r (e) λ(de) 2 + E γ i (r, X t,x r , e) U i;t,x r (e) -k U i;t,x r (e)1 {|e|≥ 1 k } λ(de) 2 ≤ Cρ( X t,x r -k X t,x r 2 ) E U i;t,x r (e) 2 λ(de)
+C E (1 ∧ |e|) 2 λ(de) × E U i;t,x r (e) -k U i;t,x r (e)1 {|e|≥ 1 k } 2 λ(de), (4.42)
By using the majorations obtain in (4.41) and in (4.42) and Cauchy-Schwartz inequality;

E Y t,x s -k Y t,x s 2 + T 0 Z t,x s -k Z t,x s 2 + E U t,x s -k U t,x s 1 {|e|≥ 1 k } 2 λ(de) ds ≤ E g(X t,x T ) -g( k X t,x T ) 2 + C ǫ E T s Y t,x r -k Y t,x r 2 + T s ρ(E X t,x r -k X t,x r 2 ) dr +C E sup s≤r≤T ρ 2 ( X t,x r -k X t,x r 2 ) × E   T s E U i;t,x r (e) 2 λ(de) dr 2   +CǫE T t E U t,x s -k U t,x s 1 {|e|≥ 1 k } 2 λ(de) ds (4.43)
For Cǫ < 1 and the Gronwall's lemma going through the dominated convergence theorem, the continuity of g and ρ and the lemma 4.1, then

E Y t,x s -k Y t,x s 2 -→ 0 k→+∞ (4.44) and in taking s = t we obtain u k (t, x) -→ u(t, x) k→+∞ . As (t, x) ∈ [0, T ] × R k is arbitrary then u k -→ u k→+∞ pointwisely.
Taking the same arguments as when getting (4.44); we once again add Lebesgue's dominated convergence theorem to have,

E T t E U t,x s -k U t,x s 1 {|e|≥ 1 k } 2 λ(de) ds -→ 0 k→+∞ . (4.45)
Step 3: Conclusion

First note that by (4.31) and the pointwise convergence of

(u k ) k to u, if (x k ) k is a sequence of R k which converges to x then ((u k (t, x k )) k ) converges to u(t, x).
Now let us consider a subsequence which we still denote by {k} such that sup s≤T X t,x s -k X t,x s 2 -→ 0 k→+∞ , P-a.s. (and then

X t,x s--k X t,x s- -→ 0 k→+∞ since X t,x s--k X t,x s-≤ sup s≤T X t,x s -k X t,x s
2 ). By (4.34), this subsequence exists. As the mapping x → β(t, x, e) is Lipschitz then the sequence

k U t,x s (e)1 {|e|≥ 1 k } k = (u k i (s, k X t,x s-+ β(s, k X t,x s-, e)) -u k i (s, k X t,x s-))1 {|e|≥ 1 k } k≥1 -→ k→+∞ (u i (s, X t,x s-+ β(s, X t,x s-, e)) -u i (s, X t,x s-)), dP ⊗ ds ⊗ dλ -a.e. on Ω × [t, T ] × E (4.46)
for any i = 1, . . . , m. Finally from (4.45) we deduce that

U t,x s (e) = (u i (s, X t,x s-+ β(s, X t,x s-, e)) -u i (s, X t,x s-)), on Ω × [t, T ] × E (4.47)
which is the desired result.

The main result

Unlike Barles and al. [START_REF] Barles | Backward stochastic differential equations and integralpartial differential equations[END_REF] our result on viscosity solutions is established for the following definition. (i) ∀x ∈ R k , u i (x, T ) ≤ g i (x) (resp. u i (x, T ) ≥ g i (x));

(ii) For any (t, x) ∈ [0, T ] × R k and any function φ of class C 1,2 ([0, T ] × R k ) such that (t, x) is a global maximum point of u iφ (resp. global minimum point of u iφ) and (u iφ)(t, x) = 0 one has

-∂ t φ(t, x) -L X φ(t, x) -h i (t, x, (u j (t, x)) j=1,m , σ ⊤ (t, x))D x φ(t, x), B i u i (t, x)) ≤ 0 (5.48) (resp.
-∂ t φ(t, x) -L X φ(t, x)h i (t, x, (u j (t, x)) j=1,m , σ ⊤ (t, x))D x φ(t, x), B i u i (t, x)) ≥ 0 .

(5.49)

The family u = (u i ) i=1,m is a viscosity solution of (1.1) if it is both a viscosity sub-solution and viscosity super-solution.

Note that

L X φ(t, x) = b(t, x) ⊤ D x φ(t, x) + 1 2 Tr(σσ ⊤ (t, x)D 2 xx φ(t, x)) + Kφ(t, x); where Kφ(t, x) = E (φ(t, x + β(t, x, e)) -φ(t, x) -β(t, x, e) ⊤ D x φ(t, x))λ(de).
                   (i) Y t,x := (Y i;t,x ) i=1,m ∈ S 2 (R m ), Z t,x := (Z i;t,x ) i=1,m ∈ H 2 (R m×d ), U t,x := (U i;t,x ) i=1,m ∈ H 2 (L 2 m (λ)); (ii) Y i;t,x s = g i (X t,x T ) - T s Z i;t,x dB r - T s E U i;t,x
r (e) µ(dr, de).

+ T s h (i) (r, X t,x r , Y i;t,x r , Z i;t,x r , E γ i (t, X t,x
r , e){(u i (t, X t,x r-+ β(t, X t,x r-, e))u i (t, X t,x r-))} λ(de))dr that fact for any

(iii) Y i;t,x T = g i (X t,x T ). ( 5
(t, x) ∈ [0, T ] × R k , ∀s ∈ [t, T ], Y i;t,x s = u i (s, X t,x s ).
Such that by the same proposition, the family (u i ) i=1,m is a viscosity solution of the following system:

       -∂ t u i (t, x) -b(t, x) ⊤ D x u i (t, x) -1 2 Tr(σσ ⊤ (t, x)D 2 xx u i (t, x)) -K i u i (t, x) -h (i) (t, x, (u j (t, x)) j=1,m , (σ ⊤ D x u i )(t, x), B i u i (t, x)) = 0, (t, x) ∈ [0, T ] × R k ; u i (T, x) = g i (x).
(5.51)

Now we have the family (u i ) i=1,m is a viscosity solution, our main objective is to found relation between (u i ) i=1,m and (u i ) i=1,m which is defined in (3.13).

For this, let us consider the system of BSDE with jumps

                       (i) Y t,x := (Y i;t,x ) i=1,m ∈ S 2 (R m ), Z t,x := (Z i;t,x ) i=1,m ∈ H 2 (R m×d ), U t,x := (U i;t,x ) i=1,m ∈ H 2 (L 2 m (λ)); (ii) Y i;t,x s = g i (X t,x T ) - T s Z i;t,x dB r - T s E U i;t,x
r (e) µ(dr, de).

+ T s h (i) (r, X t,x r , Y i;t,x r , Z i;t,x r , E γ i (t, X t,x
r , e)U i;t,x r (e) λ(de))dr;

(iii) Y i;t,x T = g(X t,x T ).

(5.52)

By uniqueness of the solution of the BSDEs with jumps (5.49), that for any s ∈ [t, T ] and ∀i ∈ {1 . . . , m},

Y i;t,x s = Y i;t,x s .
Therefore u i = u i , such that by (4.45) we obtain U t,x s (e) = (u i (s, X t,x s-+β(s, X t,x s-, e))-u i (s, X t,x s-)), on Ω× [t, T ] × E, which give the viscosity solution in the sense of definition 5.1 (see [START_REF] Hamadène | Viscosity solutions of second order integral-partial differential equations without monotonicity condition: A new result[END_REF]) by pluging (4.46) in h (i) of (5.51).

Step 2: Uniqueness For uniqueness, let (u i ) i=1,m be another family of M which is solution viscosity of the system (1.1) in the sense of definition 5.1 and we consider BSDE with jumps defined with u

i .                    (i) Y t,x := (Y i;t,x ) i=1,m ∈ S 2 (R m ), Z t,x := (Z i;t,x ) i=1,m ∈ H 2 (R m×d ), U t,x := (U i;t,x ) i=1,m ∈ H 2 (L 2 m (λ)); (ii) Y i;t,x s = g i (X t,x T ) - T s Z i;t,x dB r - T s E U i;t,x r
(e) µ(dr, de).

+ T s h (i) (r, X t,x r , Y i;t,x r , Z i;t,x r , E γ i (t, X t,x
r , e)(u i (s, X t,x s-+ β(s, X t,x s-, e))u i (s, X t,x s-)) λ(de))dr;

(iii) Y i;t,x T = g(X t,x T ).

(

) ∈ [0, T ] × R k , ∀s ∈ [t, T ], Y i;t,x s = v i (s, X t,x s ).
Such that by the same proposition, the family (v i ) i=1,m is a viscosity solution of the following system:

       -∂ t v i (t, x) -b(t, x) ⊤ D x v i (t, x) -1 2 Tr(σσ ⊤ (t, x)D 2 xx v i (t, x)) -K i v i (t, x) -h (i) (t, x, (v j (t, x)) j=1,m , (σ ⊤ D x v i )(t, x), B i u i (t, x)) = 0, (t, x) ∈ [0, T ] × R k ; u i (T, x) = g i (x).
(5.54)

By uniqueness of solution of (5.52) u i is viscosity solution of (5.53); and by proposition 3. (e) = (v i (s, X t,x s-+ β(s, X t,x s-, e))v i (s, X t,x s-)) = (u i (s, X t,x s-+ β(s, X t,x s-, e))u i (s, X t,x s-)).

(5.55) By Remark 3.4 in [START_REF] Hamadène | Viscosity solutions of second order integral-partial differential equations without monotonicity condition: A new result[END_REF]; let us considere (x k ) k≥1 a sequence of R k which converges to x ∈ R k and the two following BSDE with jumps (adaptation is w.r.t.

F k ):                            (i) Y k,t,x := (Y i;k,t,x ) i=1,m ∈ S 2 (R m ), Z k,t,x := (Z i;k,t,x ) i=1,m ∈ H 2 (R m×d ), U k,t,x := (U i;k,t,x ) i=1,m ∈ H 2 (L 2 m (λ)); (ii) Y i;k,t,x s = g i (X k,t,x T ) - T s Z i;k,t,x dB r - T s E U i;k,t,x r (e) µ(dr, de) + T s h (i) r, X k,t,x r , Y i;k,t,x r , Z i;k,t,x r , E γ i (t, X k,t,x k r , e)(u i (s, X k,t,x k s- + β(s, X k,t,x k s- , e)) -u i (s, X k,t,x k s-
)) λ(de) dr; (e) = (v i (s, k X t,x s-+ β(s, k X t,x s-, e))v i (s, k X t,x s-)), ds ⊗ dP ⊗ dλ k -a.e on [t, T ] × Ω × E; (ii) the sequence ((v k i=1,m )) k≥1 converges to v i (t, x) by using (4.44). So that x k -→ k x we take the following estimation which is obtaining by Ito's formula and by the properties of h (i) .

(iii) Y i;k,t,x k T = g(X k,t,x k T ). ( 5
E Y k,t,x k s -Y k,t,x s 2 + T 0 Z k,t,x k s -Z k,t,x s 2 + E U k,t,x k s -U k,t,x s 2 λ k (de) ds ≤ E g( k X t,x k T ) -g( k X t,x T ) 2 + 2 T s < Y k,t,x k r -Y k,t,x r
, k ∆h (i) (r) > dr

Using the same arguments as in proof (4.33), it follows that for s = t; ∀i = 1, . . . , m,

v k i (t, x k ) -→ k v k i (t, x).
Therefore by (i)-(ii) we have, for any i = 1, . . . , m, U i;t,x s (e) = (v i (s, X t,x s-+β(s, X t,x s-, e))-v i (s, X t,x s-)) ds⊗dP⊗dλ-a.e. in [t, T ]×Ω×E, ∀i ∈ {1, . . . , m}.

(5.58)

By this result we can replace (u i (s, X t,x s-+ β(s, X t,x s-, e))u i (s, X With the uniqueness of solution (5.53), we have u i = u i = v i which means that the solution of (1.1) in the sense of Definition 5.1 is unique inside the class M.

In this paper we have shown the existence and uniqueness of viscosity solution through BSDE by weakening the condition on the generator used in [START_REF] Hamadène | Viscosity solutions of second order integral-partial differential equations without monotonicity condition: A new result[END_REF].

The question that arose now is to have the existence and uniqueness of viscosity solution through the RBSDEs.

  .26) Proof. Step 1: Truncation of the Lévy measure For any k ≥ 1, let us first introduce a new Poisson random measure µ k (obtained from the truncation of µ) and its associated compensator ν k as follows:

  let η ∈ [0, T ]. Since |a + b + c| 2 ≤ 3(|a| 2 + |b| 2 + |c| 2 ) for any real constants a, b and c and by the Cauchy-Schwartz and Burkholder-Davis-Gundy inequalities we have:

Definition 5 . 1

 51 We say that a family of deterministics functions u = (u i ) i=1,m which belongs to M ∀i ∈ {1, . . . , m} is a viscosity sub-solution (resp. super-solution) of the IPDE (1.1) if:

Theorem 5 . 2

 52 Under assumptions (H1) and (H2), the IPDE (1.1) has unique solution which is the m-tuple of functions (u i ) i=1,m defined in proposition 3.3 by (3.13).Proof. Step 1: ExistenceAssume that assumptions (H1) and (H2) are fulfilled, then the following multi-dimensional BSDEs with jumps

  .50) has unique solution (Y , Z, U ). Next as for any i = 1, . . . , m, u i belongs to M, then by proposition 3.3 the (3.13), there exists a family of deterministics continuous functions of polynomial growth (u i ) i=1,m

  ;k,t,x k ) i=1,m ∈ S 2 (R m ), Z k,t,x k := (Z i;k,t,x k ) i=1,m ∈ H 2 (R m×d ), U k,t,x k := (U i;k,t,x k ) i=1,m ∈ H 2 (L 2By proof of step 2 of proposition 4.3,(Y i;k,t,x , Z i;k,t,x , U i;k,t,x 1 {|e|≥ 1 k } ) k converge to (Y i;t,x , Z i;t,x , U i;t,x ) in S 2 (R) × H 2 (R κ×d ) × H 2 (L 2 (λ)).)) k≥1 be the sequence of continuous deterministics functions such that for any t ≤ T and s ∈ [t, T ],

	and
	                                	(i) Y (ii) Y k,t,x k i;k,t,x k := (Y s = g i (X k,t,x k T + T s h (i) r, X k,t,x k ) -r , Y γ i (t, X k,t,x k T s Z i;k,t,x k dB r -m (λ)); s i;k,t,x k r , Z i;k,t,x k r , r , e)(u i (s, X k,t,x k T E s-E (iii) Y i;k,t,x k T = g(X k,t,x k T ).	U + β(s, X k,t,x k i;k,t,x k (e) µ(dr, de) r s-, e)) -u i (s, X k,t,x k s-	)) λ(de) dr;
						(5.57)
	Let ((v k i=1,m Y	i;k,t,x s	= v k i (s, k X t,x s ) and Y	i;k,t,x k s	= v k i (s, k X t,x k
						.56)

is ) ∀i = 1, . . . , m.

Such that we have respectively by proof of proposition 4.3 in step 1 and step 2:

(i) U i;k,t,x s

  ) i=1,m ∈ S 2 (R m ), Z t,x := (Z i;t,x ) i=1,m ∈ H 2 (R m×d ), U t,x := (U i;t,x ) i=1,m ∈ H 2 (L 2 m (λ));

	that (Y	t,x , Z	t,x , U	t,x ) verifies: ∀i ∈ {1, . . . , m}
	                  	(i) Y i;t,x (ii) Y t,x := (Y i;t,x s = g i (X t,x T ) -+ T s h (i) (r, X t,x r , Y T s i;t,x Z i;t,x r r , Z (iii) Y i;t,x T = g i (X t,x T ).	dB r -i;t,x r , E	T γ i (r, X t,x s U i;t,x r E r , e)U (e) µ(dr, de). i;t,x λ(de))dr; r
					(5.59)
	It follows that
					∀i ∈ {1, . . . , m}, Y	i;t,x = Y i;t,x .
					t,x s-)) by U	i;t,x s	(e) in (5.55), we deduce

Appendix

Barles and al.'s definition for viscosity solution of IPDE (1.1) Definition 5. [START_REF] Hamadène | Viscosity solutions of second order integral-partial differential equations without monotonicity condition: A new result[END_REF] We say that a family of deterministics functions u = (u i ) i=1,m which is continuous ∀i ∈ {1, . . . , m}, is a viscosity sub-solution (resp. super-solution) of the IPDE (1.1) if:

The family u = (u i ) i=1,m is a viscosity solution of (1.1) if it is both a viscosity sub-solution and viscosity super-solution.

Another Mao condition

In this paper it was mainly a question of the p-order Mao condition, it is necessary to know that there exists another condition of Mao which is mainly used in the case of monotony of the generator for apply the comparison theorem at the viscosity solution.

Definition 5.4 f satisfies the p-order one-sided Mao condition in x i.e., there exists a nondecreasing, concave function ρ(•) : R + → R + with ρ(0) = 0, ρ(u) > 0, for u > 0 and

Remark 2 Applying Cauchy-Schwartz inequality for the p-order one-sided Mao condition, we deduce the p-order Mao condition.