François Cury

Philippe Sarrazin

Aïna Chalabaev
email: aina.chalabaev@unice.fr

Brenda Major

Barbara François Cury

Physiological markers of challenge and threat mediate the effects of performance-based goals on performance

Keywords: achievement goals, avoidance, approach, challenge, threat, cardiovascular responses, performance

come

Physiological markers of challenge and threat mediate the effects of performance-based goals on performance

Aïna Chalabaev, Brenda Major,

ACCEPTED MANUSCRIPT

Achievement goals and psychophysiology 3 Physiological markers of challenge and threat mediate the effects of performance-based goals on performance Why do some students strive for success whereas others give up from even the smallest difficulty? Over the past decades, the achievement goal perspective [START_REF] Elliot | A conceptual history of the achievement goal construct[END_REF] has attempted to answer this question. According to this theory, the goals that people adopt within achievement situations create a framework for how they interpret, evaluate, and act on achievement-relevant information. For example, some people adopt a performance-approach goal, trying to perform well compared to others, whereas others adopt a performance-avoidance goal, trying to avoid performing poorly relative to others [START_REF] Elliot | A hierarchical model of approach and avoidance achievement motivation[END_REF]. Adopting a performance-approach goal is associated with more adaptive achievement outcomes than adopting a performance-avoidance goal, including lower anxiety, higher task absorption, more intrinsic motivation and better performance (e.g., [START_REF] Cury | The trichotomous achievement goal model and intrinsic motivation: A sequential mediational analysis[END_REF][START_REF] Elliot | Approach and avoidance achievement goals and intrinsic motivation: A mediational analysis[END_REF]; for a review see [START_REF] Elliot | A conceptual history of the achievement goal construct[END_REF].

Why do these goals have such different implications for achievement? One possibility is that type of goal influences the ways in which people appraise achievement tasks, i.e., assess the demands of the task and their resources for meeting those demands (Lazarus & Folkman, 1984). According to [START_REF] Elliot | Approach and avoidance achievement goals and intrinsic motivation: A mediational analysis[END_REF], in a performance-approach goal orientation, individuals perceive the achievement setting as a challenge, i.e. they appraise their resources as exceeding task demands. In contrast, in a performance-avoidance goal orientation, individuals construe the achievement setting as a threat, because they perceive the demands of the task as outweighing their resources [START_REF] Blascovich | The biopsychosocial model of arousal regulation[END_REF]Lazarus & Folkman, 1984). Indeed, focusing on avoiding demonstrating one's incompetence may pose a threat to self-esteem, thus increasing the psychological danger (demands) associated with a task compared to focusing on demonstrating one's competence.

Challenge and threat appraisals are key constructs because they are supposed to explain why performance-approach and avoidance goals have different consequences on achievement. However, to date, no research has tested whether these appraisals mediate the differential effects of performance-based goals on performance. This was the aim of the current research.

Using self-report measures, [START_REF] Elliot | Approach and avoidance achievement goals and intrinsic motivation: A mediational analysis[END_REF] found that participants who endorsed a performance-approach goal (relative to a performanceavoidance goal) appraised an achievement task as a more of a challenge, although performance-based goals were unrelated to threat appraisals. Nevertheless, these appraisals were not tested as mediators. One reason that may explain these results is that challenge and threat may be difficult to assess via self-reports. Thus, the current research used physiological markers of challenge and threat states identified and validated in prior research (see [START_REF] Blascovich | Challenge and threat[END_REF] for a review) in addition to self-reports. Physiological measures have several methodological advantages over self-reports; most notably, as they can be continuously and covertly recorded on line, they are less susceptible to effects of reappraisal and discounting and to self-presentational biases that can occur when one's abilities are in question. [START_REF] Elliot | Color and psychological functioning: The effect of red on performance attainment[END_REF] demonstrated the utility of using physiological measures to study achievement motivation processes. They showed that activating avoidance motivation using a nonconscious prime (the color red) affected both performance and a physiological measure of avoidance motivation (cortical activation).

According to the biopsychosocial model (BPSM) of challenge and threat (e.g., [START_REF] Blascovich | The biopsychosocial model of arousal regulation[END_REF][START_REF] Blascovich | Challenge and threat appraisals: The role of affective cues[END_REF], within motivated performance situations, four measures of cardiovascular reactivity (CVR) can be used to differentiate challenge and threat states: heart rate (HR), ventricular contractility (VC; an index of the left ventricle's contractile force), cardiac output (CO; the amount of blood in liters pumped by the heart per minute), and total peripheral resistance (TPR; an index of net constriction vs. dilation in the arterial system). Challenge is indexed by an increase in CO and a decrease in TPR, indicating increased cardiac efficiency and vasodilation in the arterioles, which provides greater blood flow to the periphery. In contrast, threat is indexed by little or no change in CO and no change or an increase in TPR, indicating less efficient cardiac output and vasoconstriction. Task engagement is common to both challenge and threat states and is indexed by an increase in HR and VC from baseline. A number of studies have provided correlational, experimental, and predictive support for these differential patterns of CVR as measures of challenge and threat appraisals (see [START_REF] Blascovich | Challenge and threat[END_REF]. For example, Blascovich, Seery, Mugridge, Weisbugh and Norris (2004) showed that baseball players who exhibited a challenge pattern of CVR while giving a baseball-related speech performed significantly better on the field in the six months following the experiment compared to those who exhibited a threat pattern of CVR during the speech.

Integrating the BPSM with achievement goal theory provides a novel way of examining the role of appraisals in performance-based goals processes. Based on the above literature, we generated the following predictions. First, we hypothesized that individuals induced to adopt a performance-approach goal would perform better on a problem-solving task than would individuals induced to adopt a performance-avoidance goal. Second, we predicted that the former would display a pattern of CVR indicative of challenge whereas the latter would display a pattern of CVR indicative of threat. We also predicted that the former would report feeling more challenged and less threatened than the latter. Third, we predicted that the effect of performance goal induction on task performance would be mediated by challenge and threat states as indexed by patterns of CVR and self-reports.

Method

Participants

Twenty-seven female undergraduates at the University of California took part in this study for course credit or $10.

Procedure and Measures

During a prescreening survey participants reported their score on the quantitative section of the Scholastic Aptitude Test (SAT). Upon arriving individually at the laboratory, a female experimenter blind to hypotheses and condition applied CV sensors to participants, who sat in an upright chair facing a computer monitor at eye level.

Participants were asked to sit quietly for five minutes, during which baseline CV data were recorded.

CV measures were recorded noninvasively and continuously via impedance cardiography (Minnesota Impedance Cardiograph, Model 304B), electrocardiography (Coulbourn ECG amplifier/coupler, Model S75-11), and continuous blood pressure (Cortronics, Model 7000) recording equipment (for more details see [START_REF] Blascovich | Predicting athletic performance from cardiovascular indexes of challenge and threat[END_REF]. Following the rest period, tape-recorded instructions informed participants that the research sought to understand psychophysiological processes underlying differences in problem-solving ability between individuals and reminded them that problem-solving skills are crucial to performance in many important subjects in college. Participants were then randomly assigned to one of two performance-based goal conditions, using an instructional manipulation based on [START_REF] Brown | A burden of proof: Stereotype relevance and gender differences in math performance[END_REF]. Instructions for the performance-avoidance condition follow, with performance-approach goal instructions shown in parentheses.

"The particular test you will be taking today is designed to help us identify people who are exceptionally weak (strong) in their problem-solving reasoning abilities. Your performance on this test will not be scored like most normal tests, but rather will be classified as either above or below a predetermined cutoff score.

If you score below (above) the cutoff score, this suggests that you are exceptionally weak (strong)-in other words, well below (above) average in your problem-solving reasoning abilities. However, scoring above (below) the cutoff score tells us little or nothing about your problem-solving abilities and potential.

You may be below average, average, or even above average. Thus, this test and the scoring method used with it are designed only to separate those who are especially weak (strong) from everyone else."

Because normative comparison is an important component of performance-based goals (e.g., [START_REF] Elliot | Achievement goals and the hierarchicalmodel of achievement motivation[END_REF], participants were also told that at the end of the experiment, they would be provided with information regarding how they did on the problem-solving test compared to other university students.

Participants next completed a manipulation check (indicating whether the purpose of the task was to select students who were exceptionally strong or weak) and indicated on a 1 to 7 scale the extent to which they felt "challenged" and "worried" as they anticipated doing the task (e.g., [START_REF] Elliot | Approach and avoidance achievement goals and intrinsic motivation: A mediational analysis[END_REF].

They then performed the problem-solving task. This was composed of 10 multiple-choice word problems with 5 answer choices, taken from a Graduate Record Examination practice book. Problems were presented on the computer monitor in front of the participants. In order to prevent movement artefacts caused by writing during the test that could interfere with measuring equipment participants had to solve the problems mentally and verbalize their responses. The computer screen automatically advanced to the next problem after one minute had passed. Performance was scored using the standard formula for scoring GRE: correct items received one point and incorrect items received a deduction of one point divided by the number of response options for that item (i.e., 5)the correction factor for guessing. Following task completion, participants were thanked and fully debriefed.

Results

Preliminary analyses

We first analyzed the manipulation checks to establish that the manipulation was perceived as intended. It was. Following standard practices, we then computed reactivity values for CV responses by subtracting the average value of the last baseline minute from the mean of the first three minutes of the task. Only the first task minutes were analyzed because CV responses have been shown to peak in the first minutes of the task [START_REF] Blascovich | The biopsychosocial model of arousal regulation[END_REF]. We first checked that participants were engaged in the task by testing HR and VC reactivity against zero. Two-tailed t tests showed significant differences from zero in HR reactivity, t(25) = 4.70, p < .001, and VC reactivity, t(25) = 3.99, p < .001, confirming that participants were engaged during the first minutes of the task.

Primary analyses

Effects of goal on task performance. To test our first prediction, we conducted a multiple regression analysis predicting task performance from goal type (we dummycoded performance-approach goal as 1 and performance-avoidance goal as 0). This analysis controlled for SAT scores in order to test the impact of performance-based goals independently of objective ability constructs (e.g., [START_REF] Elliot | Test anxiety and the hierarchical model of approach and avoidance achievement motivation[END_REF]. As predicted, performance-approach goal participants (M = 5.87) performed significantly better than performance-avoidance goal participants (M = 4.19), = .37, p = .02.

Effects of goal on physiological responses.

To test our second prediction, we performed multiple regression analyses predicting CO and TPR reactivity from goal type, controlling for SAT and baseline measures of CO or TPR. As predicted, performanceapproach goal participants showed significantly higher levels of CO reactivity, = .53, p = .01, and significantly lower levels of TPR reactivity, = -.56, p = .005, than performance-avoidance goal participants. As shown in Figure 1, performance-approach goal participants exhibited a CVR pattern consistent with a psychological state of challenge, whereas performance-avoidance goal participants exhibited a CVR pattern consistent with a psychological state of threat.

Effects of goal on self-reports. We performed similar regression analyses examining the effects of goal type on self-reported feelings of challenge and threat, controlling for SAT scores. Performance-approach goal participants reported feeling significantly more challenged prior to the task than did performance-avoidance goal participants, = .56, p < .01. Analysis of the threat item "I feel worried" revealed no effect of goal type.

Mediational analyses. To test our third prediction, we conducted meditational analyses following procedures recommended by [START_REF] Baron | The moderator-mediator variable distinction in social psychological research: Conceptual, strategic, and statistical considerations[END_REF]. First, we regressed task performance on Goal Type, SAT, CO reactivity and baseline CO.

Consistent with predictions, CO reactivity was positively related to performance, = .53, p < .001. In addition, inclusion of CO reduced the beta for Goal Type to nonsignificance ( = .12); a Sobel test showed that this reduction in magnitude of effect was significant (Z = 2.34, p = .02) (see Figure 2). Next, we regressed task performance on Goal Type, SAT, TPR reactivity and baseline TPR. TPR reactivity was negatively related to performance, = -.36, p = .04. Furthermore, inclusion of TPR reactivity in the model reduced the beta for Goal Type to nonsignificance ( =.21); the Sobel test was marginally significant, Z = 1.74, p = .08.

Finally, we regressed task performance on Goal Type, SAT, and self-reported feelings of challenge. Self-reported challenge was unrelated to performance, = -.12 and did not qualify as a mediator of the effects of goal on task performance.

Discussion

Researchers have speculated that performance-approach goals may lead people to experience achievement situations as a challenge whereas performance-avoidance goals may lead people to experience achievement situations as a threat [START_REF] Elliot | Approach and avoidance achievement goals and intrinsic motivation: A mediational analysis[END_REF]. Further, these differential states are thought to potentially mediate effects of goals on performance. The current study was the first to directly test this hypothesis by articulating the achievement goal theory with the BPSM of challenge and threat.

Collectively, results indicated that threat and challenge states, as measured by patterns of CVR but not by self-report, mediated the effect of induced goal type on performance.

Consistent with prior research (see [START_REF] Elliot | A conceptual history of the achievement goal construct[END_REF], participants induced to adopt a performance-approach goal performed better on a problem-solving task than those induced to adopt a performance-avoidance goal. Inducing these different performancebased goals also led to different patterns of CVR. Specifically, participants pursuing a performance-approach goal exhibited a physiological pattern characteristic of challenge during the task whereas participants pursuing a performance-avoidance goal exhibited a physiological pattern characteristic of threat. Most importantly, we found that these psychological states, as indexed by physiology, mediated the effect of achievement goals on task performance. These results are, to our knowledge, the first empirical evidence do demonstrate this. Similar to [START_REF] Elliot | Approach and avoidance achievement goals and intrinsic motivation: A mediational analysis[END_REF], we also found that inducing a performance-approach goal led to greater self-reported feelings of challenge, whereas inducing a performance-avoidance goal did not lead to greater self-reported feelings of threat. Self-reported challenge, however, did not mediate the effects of goal type on performance. Challenge and threat states may be difficult to assess via self-reports, however, either because participants are unable or unwilling to report on these assessing these states [START_REF] Blascovich | Predicting athletic performance from cardiovascular indexes of challenge and threat[END_REF].

One limitation of this study is the lack of a control group in which no achievement goal was induced. Consequently, we do not know if the approach goal induction had facilitative effects or if the avoidance goal induction had debilitating effects. Although characteristic of many studies in this area (e.g., [START_REF] Cury | The trichotomous achievement goal model and intrinsic motivation: A sequential mediational analysis[END_REF][START_REF] Elliot | Achievement goals, selfhandicapping, and performance attainment: A mediational analysis[END_REF][START_REF] Elliot | Approach and avoidance achievement goals and intrinsic motivation: A mediational analysis[END_REF][START_REF] Senko | Achievement goals, performance and task interest: Why perceived difficulty matters[END_REF], this should be addressed in future research.

In conclusion, this study is the first to demonstrate that challenge and threat psychological states, indexed by physiological patterns of CVR, are not only differentially induced by performance-approach vs. performance-avoidance goals, respectively, but also mediate the effects of performance-based goal on performance.

Figure 1 .

 1 Figure 1. Adjusted CO and TPR reactivity means by goal type during task performance.

Figure 2 .

 2 Figure 2. Mediation of goal type effect on performance by cardiovascular responses.

This research was supported by a Marie Curie Outgoing International Fellowship from the European Commission.


Marek J Śmieta Ński
email: smietan@math.uni.lodz.pl

An approximate Newton method for solving nonsmooth equations with infinite max functions

Keywords: nonsmooth equation, semismooth function, weak consistently approximated Jacobian, difference approximation, superlinear convergence Nonsmooth equation, Semismooth function, Weak consistently approximated Jacobian, Difference approximation, Superlinear convergence. 2000 Mathematics Subject classification codes: 65H10, 90C33

An approximate Newton method for solving nonsmooth equations with infinite max functions

Marek J.

Introduction

Many important practical problems of mathematical programming require the solving of nonlinear system of equations

F (x) = 0, (1)
where F : R n → R n is locally Lipschitz. The generalized Jacobian method for solving such systems was proposed by Qi and Sun in [START_REF] Qi | A nonsmooth version of Newton's method[END_REF] in the form

x (k+1) = x (k) -V -1 k F (x (k) ), V k ∈ ∂F (x (k) ), (2)
where ∂F (x (k) ) is the generalized Jacobian of the function F at x (k) , defined by Clarke in [START_REF] Clarke | Optimization and nonsmooth analysis[END_REF], and a matrix V k is taken arbitrarily from ∂F (x (k) ). The iteration generated by [START_REF] Chen | Superlinear convergence of smoothing quasi-Newton methods for nonsmooth equations[END_REF] is locally superlinearly convergent under the assumption of semismoothness of the function F . Xu and Chang in [START_REF] Xu | Approximate Newton methods for nonsmooth equations[END_REF] expanded method [START_REF] Chen | Superlinear convergence of smoothing quasi-Newton methods for nonsmooth equations[END_REF], replacing an iteration matrix with an adequately defined consistently approximated Jacobian, to avoid the complicated evaluations of V k . Xu and Chang proved the superlinear convergence of the method under the assumption of semismoothness. Other versions of the difference approximation were proposed also by Śmietański in [START_REF] Śmietański | New versions of approximate Newton method for solving nonsmooth equations[END_REF].

In this paper we present the generalized Newton method for solving some new class of nonlinear equations. We construct the difference approximation method for solving problems with functions defined by the operator infinite maximum, i.e. the considered function F has the following form

F (x) = max {f 1 (x), f 2 (x), ...} ,
where f p : R n → R n , p = 1, 2, ... and 'max' denotes the componentwise maximum operator. We obtain again that the method is locally superlinearly convergent under the assumption of semismoothness of F . The iterations are based on some approximation of a weak consistently approximated Jacobian, which was introduced by Śmietański in [START_REF] Śmietański | An approximate Newton method for nonsmooth equations with finite max functions[END_REF]. Earlier, Kummer in [START_REF] Kummer | Newton's method based on generalized derivatives for nonsmooth functions: convergence analysis[END_REF] presented sufficient and necessary conditions for the convergence of Newton method based on the generalized derivative (in particular the generalized Jacobian). But our approach permits to determine an approximation of generalized Jacobian in some special cases and involves the proof of superlinear convergence.

In Section 2 we give some needed good-known notions. In Section 3 we present an approximate Newton method for solving nonsmooth equations and describe a consistently approximated Jacobian, a weak consistently approximated Jacobian and their properties. In Section 4 we remind the finite difference approximation for the generalized Jacobian and propose a superlinearly convergent approximate method for solving nonsmooth equations with infinite max functions.

Notation: Throughout the paper, we use the following notation. R n denotes the vector space of n-tuples with the Euclidean norm • and L(R n ) the matrix space of n × n real matrices with the induced norm • . S(x, δ) denotes an open ball in R n with center x and radius δ, N(x, δ) is a closed ball in R l with center x and radius δ. If a j , j = 1, ..., n are vectors of R n , [a 1 , ..., a n ] will be a matrix whose j-th column is a j , and if A j , j = 1, ..., n are subsets of R n , then [A 1 , ..., A n ] will be a set of matrices in the form [a 1 , ..., a n ], where a j ∈ A for all j = 1, ..., n.

Preliminaries

Throughout this work, we assume that the functions f : R n → R and F : R n → R n are locally Lipschitz.

Let S be some set with Lebesgue's measure 0 in R n . The generalized gradient of f at x is

∂f(x) = co{lim ∇f (x (k) ) : x (k) → x, x (k) / ∈ S, x (k) ∈ D f },
where D f denotes the set of points at which f is differentiable. Clarke in [START_REF] Clarke | Generalized gradients and applications[END_REF] and [START_REF] Clarke | Optimization and nonsmooth analysis[END_REF] showed that if f is C 1 or differentiable at x and regular, then

∂f(x) = {∇f(x)}. (3)
Moreover, Clarke in [START_REF] Clarke | Optimization and nonsmooth analysis[END_REF] proved among the others that ∂f (x) is a compact subset of R n and upper semicontinuous.

The following theorem describes the generalized gradient of a function defined by means of the infinite max operator: Theorem 1 (Clarke [START_REF] Clarke | Generalized gradients and applications[END_REF], Theorem 2.1) Let U be a sequentially compact space, and let g : R n × U → R has the following properties:

(a) g(x, u) is upper semicontinuous in (x, u); (b) g is locally Lipschitz in x, uniformly for u ∈ U ; (c) g • x (x, u; h) = g ′ x (x, u; h) for each h ∈ R n ; (d) ∂ x g(x, u) is upper semicontinuous in (x, u). Then, if we let f(x) = max {g(x, u) : u ∈ U }, (i) f is locally Lipschitz; F o r P e e r R e v i e w O n l y (ii) f ′ (x; h) exists; (iii) f ′ (x; h) = f • (x; h) = max { ξ, h : ξ ∈ ∂ x g(x, u), u ∈ M(x)}, where M (x) = {u ∈ U : g(x, u) = f(x)}; (iv) ∂f(x) = co {∂ x g(x, u) : u ∈ M(x)} .
Remark: (Clarke [4], Remark after Theorem 2.1) Hypotheses (b), (c), (d) from the above theorem will follow if g is convex in x. Moreover, the conditions on g also hold when ∇ x g(x, u) exists and is continuous in (x, u). Thereby, the above theorem is a generalization of a theorem originally due to Danskin [START_REF] Danskin | The theory of max-min and its application to weapons allocation problems[END_REF].

According to Rademacher's Theorem, the local Lipschitz continuity of F implies that F is differentiable almost everywhere. Let D F be the set of points at which F is differentiable. The generalized Jacobian of F : R n → R n at x is

∂F (x) = co{lim JF (x (k) ) : x (k) → x, x (k) ∈ D F },
where JF x (k) denotes the usual Jacobian of F at x (k) . Clarke in [START_REF] Clarke | Optimization and nonsmooth analysis[END_REF]

proved that ∂F (x) ⊂ [∂F 1 (x), ..., ∂F n (x)] T , (4)
where the Cartesian product on the right side of the above inclusion denotes the set of all matrices, whose i-th row belongs to ∂F i (x) for every i. If n = 1, then ∂F (x) = ∂F 1 (x) (i.e. the generalized gradient and the generalized Jacobian coincide). Following Chen [START_REF] Chen | Superlinear convergence of smoothing quasi-Newton methods for nonsmooth equations[END_REF], we will denote the set of matrices, whose rows are the generalized gradients of components F i of the function F at x as

∂ C F (x), i.e. ∂ C F (x) = ∂F 1 (x), ..., ∂F n (x)
T . Hiriart-Urruty in [START_REF] Hiriart-Urruty | Refinements of necessary optimality conditions in nondifferentiable programming I[END_REF] stated, that the equality in (4) holds at least in two particular cases. First, when all functions F i except possibly one, e.g. F 1 , are C 1 at x, then ∂F (x) = [∂F 1 (x), ∇F 2 (x), ..., ∇F n (x)] T . The second case holds when the component functions are nondifferentiable with respect to nonrelated variables, e.g. F i is nondifferentiable only with respect to variable x i for every i = 1, ..., n.

The generalized Jacobian of F with respect to the j -th variable (the partial generalized Jacobian) at x is

∂ x j F (x) = co{lim ∇ x j F (x (k) ) : x (k) → x, x (k) ∈ D F },
where ∇ x j F x (k) denotes the vector of the usual partial derivatives with respect to the j-th variable at x (k) . From the definitions of the generalized Jacobian and the partial generalized Jacobian we may easily obtain the following inclusion:

∂F (x) ⊂ [∂ x1 F (x), ..., ∂ xn F (x)],
where the Cartesian product on the right side denotes the set of all matrices, whose i-th column belongs to ∂ x i F (x) for every i (see Hiriart-Urruty [START_REF] Hiriart-Urruty | Refinements of necessary optimality conditions in nondifferentiable programming I[END_REF] -comment to the definition of the generalized partial Jacobian after Theorem 2.1).

We will say that ∂F (x) (or Proof. Since ∂ C F (x) is nonsingular, then there exists ε > 0 sufficiently small such that ∂ C F (x) + εB is nonsingular, where B is unit ball in matrix space R n×n . By the upper semicontinuity of ∂ C F , for every ε > 0, there exists δ > 0 such that

∂ C F (x)) is nonsingular if all V ∈ ∂F (x) (V ∈ ∂ C F (x), respectively) are nonsingular.
∂ C F (y) ⊂ ∂ C F (x) + εB for all y ∈ S(x, δ).
The semismoothness was introduced for functionals by Mifflin in [START_REF] Mifflin | Semismooth and semiconvex functions in constrained optimization[END_REF]. The following definition is presented after Qi and Sun [START_REF] Qi | A nonsmooth version of Newton's method[END_REF].

F is semismooth at x if F is locally Lipschitz at x and lim V ∈∂F (x+th ′ ),h ′ →h,t↓0 V h ′ . ( 5
)
exists for any h ∈ R n . Clearly, convex functions and smooth functions are semismooth. Scalar products and sums of semismooth functions are semismooth functions (see Mifflin [START_REF] Mifflin | Semismooth and semiconvex functions in constrained optimization[END_REF]). The assumption [START_REF] Clarke | Optimization and nonsmooth analysis[END_REF] implies that the Hadamard directional derivative exists (see Qi and Sun [START_REF] Qi | A nonsmooth version of Newton's method[END_REF]):

lim h ′ →h,t↓0 F (x + th ′ ) -F (x) t = lim V ∈∂F (x+th ′ ),h ′ →h,t↓0 V h ′ .
Lemma 3 (Qi, Sun [START_REF] Qi | A nonsmooth version of Newton's method[END_REF], Theorem 2.

3) The following statements are equivalent:

(i) F is semismooth at x; (ii) V h -F ′ (x; h) = o ( h ) for V ∈ ∂F (x + h), h → 0; (iii) lim h→0 F ′ (x+h;h)-F ′ (x;h) h = 0.
Remark: Chen in [START_REF] Chen | Superlinear convergence of smoothing quasi-Newton methods for nonsmooth equations[END_REF] suggested that the statement (ii) of Lemma 3 holds also for any V ∈ ∂ C F (x + h), which is obvious because the semismoothness of F at x is equivalent to the semismoothness of each of its components F i at x, what follows by Corollary 2.4 from [START_REF] Qi | A nonsmooth version of Newton's method[END_REF] and Lemma 3(iii)).

Additionally, Qi and Sun [START_REF] Qi | A nonsmooth version of Newton's method[END_REF] remarked that if F is semismooth at x, then for any h → 0

F (x + h) -F (x) -F ′ (x; h) = o ( h ) . (6)
From ( 6) it follows that a semismooth function is B-differentiable.

Approximate Newton method

First, we introduce an approximate Newton iteration. Let us consider the version of Newton method in which the next point is computed by

x (k+1) = x (k) -J(x (k) , s (k) ) -1 F (x (k) ), (7)
where

J(x (k) , s (k) ) ∈ L(R n
) is some approximation of the generalized Jacobian of F and s (k) is an l-dimensional vector of parameters. The iteration ( 7) is a generalization of the smooth approximate Newton method, which was established by Ortega and Rheinboldt in [START_REF] Ortega | Iterative solution of nonlinear equations in several variables[END_REF]. The construction of the matrix J(x, s) is here a principal difficulty. Two versions of possible constructions of J(x, s) were presented in [START_REF] Śmietański | New versions of approximate Newton method for solving nonsmooth equations[END_REF]. Now, we present main results discussed in [START_REF] Xu | Approximate Newton methods for nonsmooth equations[END_REF] and [START_REF] Śmietański | New versions of approximate Newton method for solving nonsmooth equations[END_REF], which are necessary to prove the superlinear convergence. First, we have to remind a notion of CAJ introduced by Xu and Chang in [START_REF] Xu | Approximate Newton methods for nonsmooth equations[END_REF]:

Definition 4 Let J : D × D l ⊂ R n × R l → L(R n ). If 0 ∈ R l is a limiting point of D l , and if lim s→0 dist [J(x, s); ∂F (x)] = 0, uniformly with respect to x ∈ D, then J(x, s) is called a consistently approxi- mated Jacobian of F in D (CAJ).
Here, dist [J(x, s); ∂F (x)] = min

V ∈∂F (x) J(x, s) -V .
A simple approximate method for solving nonsmooth equations is based on a consistently approximated Jacobian.. However, to allow the solving a nonsmooth equations with function defined by the operator maximum, the weak consistently approximated Jacobian has been introduced in [START_REF] Śmietański | An approximate Newton method for nonsmooth equations with finite max functions[END_REF].

Definition 5 Let J : D × D l ⊂ R n × R l → L(R n ). If 0 ∈ R l is a limiting point of D l , and if lim s→0 dist [J(x, s); ∂ C F (x)] = 0, (8)
uniformly with respect to x ∈ D, then J(x, s) is called a weak consistently approximated Jacobian of F in D (WCAJ).

Here, again

dist[J(x, s); ∂ C F (x)] = min V ∈∂C F (x)
J(x, s) -V .

Remark: (i) Let us notice that equality [START_REF] Kummer | Newton's method based on generalized derivatives for nonsmooth functions: convergence analysis[END_REF] in the above definition is similar to the Jacobian consistence property given by Chen, Qi and Sun in [START_REF] Chen | Global and superlinear convergence of the smoothing Newton method and its application to general box constrained variational inequalities[END_REF] in the form lim

ε↓0 dist[(∇ x f (x, ε)) T ; ∂ C F (x)] = 0 for any x ∈ R n ,
where the B-subdifferential is approximated by the Jacobian of some

C 1 function f defined on R n × (0, +∞), called a smooth approximation of F . (ii) If J(x, s) is a CAJ of F in D, then J(x, s) is a WCAJ of F in D (it follows
obviously by Definitions 4, 5 and inclusion (4)).

The following lemma, theorem and corollary were presented in [START_REF] Śmietański | An approximate Newton method for nonsmooth equations with finite max functions[END_REF] to prove a convergence an approximate Newton method for nonsmooth equations with finite max function.

* ∈ R n . Suppose that F is semismooth at x * and that J(x, s) is a WCAJ of F (x) in a neighborhood of x * . Then F ′ (x * + h, h) -J(x * + h, s)h ≤ w(h, s) h ,
where lim h→0,s→0 w(h, s) = 0.

Theorem 7 Let x * ∈ D be a solution of [START_REF] Bompart | Second-order nonsmooth optimization for H ∞ synthesis[END_REF]. Suppose that F is semismooth at

x * and that ∂ W F (x * ) is nonsingular. Let J : D × D l ⊂ R n × R l → L(R n ) be a WCAJ of F in D
and assume that 0 is a limiting point of D l . Then, there exist δ > 0 and γ > 0 such that J(x, s) is nonsingular for x ∈ S (x * , δ) , s ∈ D l ∩ N (0, γ). Furthermore, the function

G(x, s) := x -J(x, s) -1 F (x) (9
)

is contractive in the sense that x * -G(x, s) ≤ u(x, s) x -x * for all x ∈ D ∩ S(x * , δ), s ∈ D l ∩ N (0, γ)
, where lim x→x * ,s→0 u(x, s) = 0.

Corollary 8 Suppose that the assumptions of Theorem 7 are satisfied. Let G(x, s) be defined by ( 9) and let g : R n → R l be a continuous function with g(x * ) = 0. If s = g(x), then there exists δ > 0 such that G(x, g(x)) is contractive in S(x * , δ). Thereby, the sequence produced by [START_REF] Hiriart-Urruty | Refinements of necessary optimality conditions in nondifferentiable programming I[END_REF] converges to x * superlinearly for a sufficiently good starting point.

Practical method for equations with max function

Now we will present some version of the difference approximation which offers a practical construction of a weak consistently approximated Jacobians for nonsmooth equations with maximum function. Clearly, the existence of a WCAJ, which ensure convergence of the method, depends on the local properties of F (e.g. semismoothness of F and nonsingularity of the generalized Jacobian of F ). First, we remind a notation for the problem. Assume that F in (1) has the following form:

F (x) = max {F 1 (x), ..., F m (x)} , (10)
where F p : R n → R n , p = 1, ..., m, and "max" denotes the componentwise maximum operator. Let I(x) denotes the active index set at x, i.e. the set {i : andA i p -the effective region of an active smooth piece of function F i , i.e. the set {x :

F (x) = f i (x)},
F i (x) = f p,i (x)}.
Remark: If all components F p are continuously differentiable, then F is a P C 1 function. Some important useful property of such functions has been studied by Pang and Ralph in [START_REF] Pang | Piecewise smoothness, local invertibility and parametric analysis of normal maps[END_REF]: for P C 1 function F , the convex hull of the set {∇f (x) : i ∈ Ĩ(x)} is equal to the generalized Jacobian of F at x, where For nonlinear equations with function F in the form [START_REF] Mifflin | On VU -theory for functions with primal-dual gradient structure[END_REF] we may construct a weak consistently approximated Jacobian in the following way:

Ĩ(x) = {i : x ∈ cl int{x ∈ V : i ∈ I(z)}} and V is a neighborhood of x.
[J(x, s)] i,j =                      F i 1 (x+sj ej )-F i 1 (x) sj for x ∈ A i 1 , ... F i p (x+sj ej )-F i p (x) sj for x ∈ A i p and x / ∈ p-1 k=1 A i k , ... F i m (x+s j e j )-F i m (x) sj for x ∈ A i m and x / ∈ m-1 k=1 A i k , (11)
where i, j ∈ {1, ..., n}, s i > 0 for j = 1, ..., n. Additionally, assume that x + s j e j ∈ A i p for suitable p. Because F is a P C 1 function, then the considered points belong to the closure of the effective region of a considered active smooth piece. Otherwise one can always construct a P C 1 function which is active at x but not active at any neighborhood of x (see Pang and Ralph [START_REF] Pang | Piecewise smoothness, local invertibility and parametric analysis of normal maps[END_REF]).

The following lemma permits to formulate the theorem about the superlinear convergence of the sequence [START_REF] Hiriart-Urruty | Refinements of necessary optimality conditions in nondifferentiable programming I[END_REF] with a WCAJ defined by [START_REF] Ortega | Iterative solution of nonlinear equations in several variables[END_REF]. Lemma 9 ( [START_REF] Śmietański | An approximate Newton method for nonsmooth equations with finite max functions[END_REF], Lemma 17) Suppose that F p , p = 1, ..., m are C 1 or differentiable and regular. Then, for J(x, s) defined by [START_REF] Ortega | Iterative solution of nonlinear equations in several variables[END_REF], the following inclusion holds:

lim s↓0 J(x, s) ∈ ∂ C F (x) . ( 12
)
Theorem 10 Let x * be a solution of [START_REF] Bompart | Second-order nonsmooth optimization for H ∞ synthesis[END_REF] with F in the form [START_REF] Mifflin | On VU -theory for functions with primal-dual gradient structure[END_REF]. Assume that F is semismooth in a neighborhood of x * and that ∂F (x * ) is nonsingular. Suppose that, for J(x, s) defined by [START_REF] Ortega | Iterative solution of nonlinear equations in several variables[END_REF], the inclusion [START_REF] Pang | Piecewise smoothness, local invertibility and parametric analysis of normal maps[END_REF] holds uniformly with respect to x in a neighborhood of x * . Let g : R n → R l be a continuous function for which g(x * ) = 0. Then, for s = g(x), there exists δ > 0 such that, if the starting point x 0 belongs to S(x * , δ), then the sequence x (k) generated by [START_REF] Hiriart-Urruty | Refinements of necessary optimality conditions in nondifferentiable programming I[END_REF] is well defined and converges to x * superlinearly.

Proof. It follows from Lemma 9 that J(x, s) is a weak consistently approximated Jacobian of funcion F in some neighborhood of x * under the assumption that (12) holds uniformly with respect to x in a neighborhood of x * for s ∈ D l ∩ N (0, γ), where γ is takie, as in Theorem 7. The rest of proof follows directly from Corollary 8.

Remark: Theorem 10 states the convergence properties of a nonsmooth version of the classical approximate Newton method (the generalized Jacobian method). The standard difference approximations are only suitable for a restricted class of problems. Therefore, in the sequel we will introduce other techniques which allow to solve more complicated problems.

F (x) = max {Φ(x, t) : t ∈ T } for x ∈ R n ,
where T is a closed subset of R m and Φ(x, t) and its first-and second-order partial derivatives with respect to components of x are continuous on R n × T . Such functions have been considered e.g. by Mifflin and Sagastizábal in [START_REF] Mifflin | On VU -theory for functions with primal-dual gradient structure[END_REF]. Moreover, there is nonsmooth infinite max-operator in optimization programs which arise in automatic control applications for H ∞ controller synthesis (see e.g. Bompart, Noll and Apkarian, [START_REF] Bompart | Second-order nonsmooth optimization for H ∞ synthesis[END_REF]). So, assume that the function F in (1) has the following form

F (x) = max y∈Y {Φ(x, y)} , ( 13
)
where Φ : R n ×Y → R n , Y is some compact set in R m , and "max", as previously, denotes the componentwise maximum operator.

For equation with infinite max it is possible to construct a difference approximation in the form

[J 1 (x, s)] i,j = Φ i (x + s j e j , y) -Φ i (x, y) s j , ( 14
)
where i, j ∈ {1, ..., n}, s j > 0 for j = 1, ..., n, and y is an arbitrarily selected element of

M i (x) = y ∈ Y : F i (x) = Φ i (x, y) . Now, A i y denotes the set x : F i (x) = Φ i (x, y) for some y ∈ Y . Assume that x + s j e j ∈ A i y for suitable y ∈ Y.
Lemma 11 Suppose that ∇ x Φ(x, y) exists and is continuous in (x, y). Then, for J 1 (x, s) defined by [START_REF] Śmietański | New versions of approximate Newton method for solving nonsmooth equations[END_REF], the following inclusion holds:

lim s↓0 J 1 (x, s) ∈ ∂ C F (x).
Proof. The approximation ( 14) is well defined, because we suppose that x + s j e j ∈ A i y for suitable y ∈ Y . Since Φ is directionally differentiable for each direction h ∈ R n , we have, for y

∈ M i (x), lim s↓0 [J 1 (x, s)] i,j = lim s↓0 Φ i (x + s j e j , y) -Φ i (x, y) s j = Φ i ′ x (x, y; e j ) = ∇ x Φ i (x, y); e j .
Since Φ i are differentiable, by remark after Theorem 1 and (3), we obtain and we obtain the desired conclusion.

∂F i (x) = co ∇ x Φ i (x, y) : y ∈ M i (x) . Then lim s↓0 [[J 1 (x, s)] i,1 , ..., [J 1 (x, s)] i,n ] = [ ∇ x Φ i (x, y); e 1 , ..., ∇ x Φ i (x, y); e n ] = ∇ x Φ i (x, y) for x ∈ A i y , y ∈ M i (x). All these facts give us that lim s↓0 [[J 1 (x, s)] i,1 , ..., [J 1 (x, s)] i,n ] ∈ ∂F i (x) for i = 1, ...,
Clearly, similarly as in finite maximum, the above lemma allows to formulate and to prove a suitable theorem about convergence of approximate Newton method [START_REF] Hiriart-Urruty | Refinements of necessary optimality conditions in nondifferentiable programming I[END_REF] based on difference approximation.

Approximation [START_REF] Śmietański | New versions of approximate Newton method for solving nonsmooth equations[END_REF] of the weak consistently approximated Jacobian uses the difference quotients based on two points (i.e. the one-sided approximation). In a similar way, it is possible to construct the approximation of a WCAJ based on three points (i.e. the central approximation), which gives better numerical results in some cases:

[J 2 (x, s)] i,j = Φ i (x + s j e j , y) -Φ i (x -s j e j , y) 2s j , (15)
where i, j ∈ {1, ..., n}, s j > 0 for j = 1, ..., n, and y is an arbitrarily selected element of M i (x) = y ∈ Y : F i (x) = Φ i (x, y) . For the above approximations we have a lemma analogous to the previous one:

Lemma 12 Suppose that ∇ x Φ(x, y) exists and is continuous in (x, y). Then, for J 2 (x, s) defined by [START_REF] Śmietański | An approximate Newton method for nonsmooth equations with finite max functions[END_REF], the following inclusion holds:

lim s↓0 J 2 (x, s) ∈ ∂ C F (x).
Proof. For every j ∈ {1, ..., n} we have Φ i (x + s j e j , y) -Φ i (xs j e j , y) 2s j = = Φ i (x + s j e j , y) -Φ i (x, y) + Φ i (x, y) -Φ i (xs j e j , y)

2s j = = 1 2
Φ i (x + s j e j , y) -Φ i (x, y) s j + Φ i (x, y) -Φ i (xs j e j , y) s j .

Since Φ is directionally differentiable for each direction h ∈ R n , we have, for y ∈ M i (x) lim s j ↓0

Φ i (x + s j e j , y) -Φ i (x, y) s j = Φ i ′ x (x, y; e j ), j ∈ {1, ..., n} . and

lim sj ↓0 Φ i (x, y) -Φ i (x -s j e j , y) s j = -Φ i ′ x (x, y; -e j ), j ∈ {1, ..., n} ,
Therefore, similarly as in the previous lemma, the differentiability of Φ i , remark after Theorem 1, (3) and a convexity of the generalized Jacobian (Proposition 2.6.2(i) in [START_REF] Clarke | Optimization and nonsmooth analysis[END_REF]) give the conclusion. The approximate Newton method presented in this paper has been implemented in DevC++ to test the numerical behavior of algorithm. All computations have been performed in double precision, which has the absolute range 5.0 • 10 -324 , 1.7 • 10 308 and gives the accuracy 15 ÷ 16 significant digits. The termination criterion was

x (k+1) -x (k) ≤ ε 1 or F x (k) ≤ ε 2 ,
where ε 1 = 1.0E -08 and ε 2 = 1.0E -12. In the tables containing the results, x (0) denotes the starting point, N it -the number of iterations needed to reach the desired precision, ∆ x -an absolute error of last approximation of solution and x is the exact solution, to which the iteration sequence was convergent. Example 1.

Consider the equation ( 1) with the following infinite max function

F (x) = max n∈N -n • sin (x/n) x .
The equation F (x) = 0 has four solutions x = kπ, k ∈ {-2, -1, 1, 2}. The problem was solved by the generalized Jacobian method for infinite max functions with a WCAJ approximated by ( 14) and [START_REF] Śmietański | An approximate Newton method for nonsmooth equations with finite max functions[END_REF]. We have tested the method with s = 1.0E -5, 1.0E -10 , |F (x)| (in the latter case s decreases to 0, similarly as in the classical Steffensen method). The Tables 1 and2 contain the results of the computations. Depending on the starting point, we notice the convergence to different solutions. For all initial points we have obtained the given solution with the precision at least 1.0E -09. Usage [START_REF] Śmietański | An approximate Newton method for nonsmooth equations with finite max functions[END_REF] allows us to start the computations far from a solution. Unfortunately, parameter s can not be too large.

If a similar equation with cosine has been solved then the results were very similar despite the fact that equation has only one solution x = π.

Example 2.

Consider the system of equations with the infinite max functions

F (x) = max n∈N {Φ(x, n)} where Φ(x, n) = n 1+n x 2 1 + 2 √ n 1+n x 2 -n |x 1 | + |x 2 | .
This equation has only one solution x * = (0, 0). We have tested the method with s = 1.0E -10 , |F (x)|. The Tables 3 and4 contain the results of the computations. If we used the approximation ( 14), we obtained the convergence to the solution almost for all tested starting points. However, a degenerate behavior appeared for some cases, especially for too large s. Using (15) we notice troubles with choice of a starting points to ensure the convergence to the solution.

11 Newton-like methods are important tools for solving nonsmooth equations. They are useful also when the evaluation of the Jacobian matrix is difficult or impossible (at least analytically) for smooth equations. As stated above, we have developed some version of approximate Newton method for solving nonsmooth equations with infinite max function. We have proved that under mild assumptions the sequence generated by the method is locally and superlinearly convergent for semismooth equations. The performance of the method is evaluated in the terms of the number of iterations required. The analysis of the numerical results seems indicate that the method is promising for solving semismooth systems because the tested equations were solved satisfactorily. The experimental results also indicate that the method has at least superlinear convergence. The detailed analysis of the results for Example 2 lets state that the convergence of method was even almost quadratic for some initial points. So, the important conclusion is that the algorithm allows us to find various solution of equation, however if there are many solutions of the problem then the relationship between the starting point and the obtained solution is unpredictable. It is easy to observe in Tables 1 and4 that choice of parameter s has not relevant effect on speed of convergence in terms of the number of iterations. In Table 3 some degenerate behavior of the method can be noticed for some initial points, if we use not sufficiently small constant parameter s in an approximation of WCAJ. Moreover, the method with the approximation based on central formula ( 14) is sensitive because to the choice of the parameter and the starting point.

F
s = 10 -5 s = 10 -10 s = |F (x)| x (0) N it ∆ x N it ∆ x x N it ∆ x x -30.0 5

6 Page 6 Lemma 6

 666 of 15 URL: http://mc.manuscriptcentral.com/gcom E-mail: ijcm@informa.Let x

  An absolute value function is a simplest example of finite max function, which arises in practice. Really |x| = max {x, -x} .

  Many of the convex function, which arises in practical problems, are maximum value functions of the form

Table 4 .

 4 0, 1.0) 4586 10 -15 looped -37 10 -16 (2.0, 2.0) 2436 10 -13 looped ---(5.0, 5.0) 3552 10 -10 looped -.0, 1.0) 669 10 -10 looped -67 10 -21 Results obtained for Example 2 using (15). s = 10 -10 s = |F (x)| x ://mc.manuscriptcentral.com/gcom E-mail: ijcm@informa.com International Journal of Computer Mathematics

Table 1 .

 1 Results obtained for Example 1 using[START_REF] Śmietański | New versions of approximate Newton method for solving nonsmooth equations[END_REF].

		s = 10 -5	s = 10 -10		s = |F (x)|
	x (0)	N it	∆ x	N it	∆ x	x	N it	∆ x	x
	-15.0	7	10 -13	7	10 -14 -2π	6	10 -14 -π
	-10.0	9	10 -14	8	10 -14 -π	7	10 -10	π
	-5.0	5	10 -13	5	10 -14 -2π	5	10 -12 -2π
	-2.0	4	10 -12	4	10 -13 -π	4	10 -13 -π
	-1.0	5	10 -9	5	10 -9	-π	5	10 -16 -2π
	-0.5	4	10 -13	4	10 -14 -2π	-	-	-
	-0.1	6	10 -12	6	10 -13	π	6	10 -14	2π
	0.1	6	10 -13	6	10 -14 -π	5	10 -16	2π
	0.5	7	10 -11	4	10 -14	2π	5	10 -9	π
	1.0	5	10 -9	5	10 -9	π	4	10 -13	π
	2.0	4	10 -12	4	10 -13	π	4	10 -14	π
	5.0	8	10 -13	5	10 -14	2π	5	10 -16	2π
	10.0	7	10 -10	-	-	2π	7	10 -12 -π
	15.0 20 10 -13	7	10 -15	2π	6	10 -15	2π

Table 2 .

 2 Results obtained for Example 1 using[START_REF] Śmietański | An approximate Newton method for nonsmooth equations with finite max functions[END_REF].