...any priori normality assumptions, such as Robinson's constraint qualification, and remain informative even for abnormal points. Moreover, our optimality conditions take into account the second-order effect of the curvature of the set in the inclusion constraints.

Key words: second-order necessary conditions of optimality; nondegeneracy; curvature effect

MSC2000 subject classification: Primary: 49K27, 90C48; secondary: 90C46, 90C34

OR/MS subject classification: Primary: Programming; secondary: infinite dimensional, nonlinear

History: Received October 16, 2004; revised February 20, 2005, and March 12, 2005.

1. Introduction. In this article, we derive second-order necessary conditions of optimality for the following abstract constrained optimization problem

f(x) [right arrow] min, x [member of] C, [F.sub.1](x) [less than or equal to] 0, [F.sub.2](x) = 0,

where X is a vector space, C [subset or equal to] X is a given closed (in some natural sense) set, f: X [right arrow] [R.sup.1], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] are given smooth mappings ([k.sub.1] and [k.sub.2] are also given positive integers). As usual, by the nonpositiveness of a finite-dimensional vector, we mean the nonpositiveness of its coordinates.

This general problem framework has been the subject of intense research (see, for example, Bonnans and Shapiro [4, 5], Bonnans et al. [6], and references therein). Note that, in Bonnans and Shapiro [4, 5] and Bonnans et al. [6], constraints of the form F(x) [member of] C, where F: X [right arrow] Y and C [subset or equal to] Y are given, were actually considered. However, the latter can be reduced to the problem setting presented above by a standard trick (see [section]4 below). This problem setting covers a wide range of classes of optimization problems such as, for example, nonlinear programming, dynamic optimization (and in particular, optimal control problems), semidefinite and semi-infinite programming, composite parameterized optimization problems, and variational problems in mechanics.

It is worth pointing out that, besides the usual inequality and equality constraints under smoothness hypotheses usually considered for second-order conditions, our problem also features constraints in the form of an inclusion into a given set that is assumed to be merely closed. Notably, we dispense with the widely used property of convexity.

One key feature that strongly departs from some well-established optimization literature, Bonnans and Shapiro [5], Cominetti [7], Rockafellar and Wets [17], and Cominetti and Penot [8], consists in the fact that our optimality conditions are obtained and remain informative without Robinson's constraint qualification condition (in other words, even for abnormal points, Arutyunov [1]).

The basic idea consists in using additional information from second-order conditions to select an appropriate subset of the set of multipliers satisfying the local necessary conditions of optimality. Our approach is based in a perturbation method developed in Arutyunov [1].

We also point out that, by taking into account the second-order effect of the curvature of the set in the inclusion constraints (see Kawasaki [12, 13], Cominetti and Penot [8]), the range of applications of our conditions matches that of the current state of the art under similar assumptions. Here, we investigate only necessary optimality conditions. Second-order sufficient conditions of optimality and their relations with necessary conditions are obtained in Bonnans and Shapiro [5] and Cominetti [7]. This article is organized as follows.

In [section]2, we state the basic version of the considered optimization problem and present our necessary conditions of optimality in the form of an extremum principle, together with some key definitions and results and a discussion of the assumptions. A detailed proof of the extremum principle and of the main auxiliary results is presented in [section]3. In [section]4, we state and prove our necessary conditions for an optimization problem exhibiting functional inclusion constraints. Both [subsection]2 and 4 also include some examples illustrating the key features and advantages of our main result. Finally, an auxiliary technical result is presented and proved in the appendix.

2. Statement of the extremum principle. Given a vector space X, a set C [subset or equal to] X, mappings [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], and a function f: X [right arrow] [E.sup.1], we consider the following optimization problem

(2.1) (P) f(x) [right arrow] min, x [member of] C, [F.sub.1](x) [less than or equal to] 0, [F.sub.2](x) = 0.

Our main goal is to obtain first- and second-order necessary extremum conditions for this problem under some assumptions about smoothness and properties of the set C. Let us introduce the notation and assumptions needed in what follows.

Let k = [k.sub.1] + [k.sub.2] and F = ([F.sub.1], [F.sub.2]): X [right arrow] Y = [R.sup.k]. We consider the so-called finite topology in the vector space X. We denote by M the set of all finite-dimensional linear subspaces M [subset or equal to] X. A set is open in the finite topology if and only if its intersection with any M [member of] M is open in the unique Hausdorff vector topology of M. (1) We denote the finite topology by [tau]. If a space X is infinite-dimensional, then equipping it with the finite topology [tau] does not, in general, turn it into a topological vector space because the addition is, as a rule, discontinuous. On the other hand, the finite topology is stronger than all topologies that transform X into a vector topological space and, therefore, a local minimum with respect to the finite topology is the weakest one among all types of minima considered.

Fix a point [x.sub.0] [member of] X. We assume that functions f and F are twice continuously differentiable in a neighborhood of [x.sub.0] with respect to the finite topology [tau]. This means that, for an arbitrary M [member of] M containing the point [x.sub.0], the restrictions of f and F to M are twice continuously differentiable in a certain (depending on M) neighborhood of [x.sub.0]. Therefore, there exist a linear functional a [member of] X* (X* is the space algebraically dual to X), a linear operator A: X [right arrow] Y, a bilinear form q: X x X [right arrow] [R.sup.1], a bilinear mapping Q: X x X [right arrow] Y, and mappings [a.sub.0]: X [right arrow] [R.sup.1] and [alpha]: X [right arrow] Y, such that, [for all] x [member of] X,

f(x) = f([x.sub.0]) + (a, x - [x.sub.0]) + 1/2q[[x - [x.sub.0]].sup.2] + [[alpha].sub.0](x - [x.sub.0]),

F(x) = F([x.sub.0]) + A(x - [x.sub.0]) + 1/2Q[[x - [x.sub.0]].sup.2] + [alpha](x - [x.sub.0]),

and, for an arbitrary M [member of] M, such that x [member of] M,

[[alpha].sub.0](x - [x.sub.0])/[[parallel]x - [x.sub.0][parallel]].sup.2.sub.M] [right arrow] and [absolute value of [alpha](x - [x.sub.0])/[[parallel]x - [x.sub.0][parallel]].sup.2.sub.M] [right arrow] as x [right arrow] [x.sub.0],

where [parallel] * [[parallel].sub.M] is a finite-dimensional norm in M.

Here and in what follows, we denote a bilinear mapping B by B[x, x] or B[[x].sup.2]. The mappings a and q are denoted, respectively, by ([differential]f/[differential]x)([x.sub.0]) and ([[differential].sup.2]f/[differential] [x.sup.2])([x.sub.0]). They are called the first- and second-order derivatives of f. A similar notation is used for the mapping F and other functions.

Regarding the set C, we assume it is closed in the finite topology.

The first-order necessary conditions for problem (P) used below are based on Mordukhovich's normal cone to the set C at [x.sub.0] [member of] C. Take any linear subspace M [member of] M such that [x.sub.0] [member of] C [intersection] M. Let us introduce Mordukhovich's normal cone Mordukhovich [16, 14] to the set C [intersection] M at [x.sub.0] denoted by [N.sup.M]([x.sub.0], C). For x [member of] M,...

NOTE: All illustrations and photos have been removed from this article.



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