Optimization Problems In Banach Spaces Research Articles

Well-posedness of a class of perturbed optimization problems in Banach spaces

Let X be a Banach space and Z a nonempty subset of X. Let J : Z → R be a lower semicontinuous function bounded from below and p ⩾ 1 . This paper is concerned with the perturbed optimization problem of finding z 0 ∈ Z such that ‖ x − z 0 ‖ p + J ( z 0 ) = inf z ∈ Z { ‖ x − z ‖ p + J ( z ) } , which is denoted by min J ( x , Z ) . The notions of the J-strictly convex with respect to Z and of the Kadec with respect to Z are introduced and used in the present paper. It is proved that if X is a Kadec Banach space with respect to Z and Z is a closed relatively boundedly weakly compact subset, then the set of all x ∈ X for which every minimizing sequence of the problem min J ( x , Z ) has a converging subsequence is a dense G δ -subset of X ∖ Z 0 , where Z 0 is the set of all points z ∈ Z such that z is a solution of the problem min J ( z , Z ) . If additionally p > 1 and X is J-strictly convex with respect to Z, then the set of all x ∈ X for which the problem min J ( x , Z ) is well-posed is a dense G δ -subset of X ∖ Z 0 .

Existence of Solutions for Nonconvex and Nonsmooth Vector Optimization Problems

We consider the weakly efficient solution for a class of nonconvex and nonsmooth vector optimization problems in Banach spaces. We show the equivalence between the nonconvex and nonsmooth vector optimization problem and the vector variational-like inequality involving set-valued mappings. We prove some existence results concerned with the weakly efficient solution for the nonconvex and nonsmooth vector optimization problems by using the equivalence and Fan-KKM theorem under some suitable conditions.

Existence of Exact Penalty and Its Stability for Nonconvex Constrained Optimization Problems in Banach Spaces

In this paper we use the penalty approach in order to study two constrained minimization problems. A penalty function is said to have the generalized exact penalty property if there is a penalty coefficient for which approximate solutions of the unconstrained penalized problem are close enough to approximate solutions of the corresponding constrained problem. In this paper we show that the generalized exact penalty property is stable under perturbations of cost functions, constraint functions and the right-hand side of constraints.

Some vector optimization problems in Banach spaces with generalized convexity

In this paper, we consider a vector optimization problem where all functions involved are defined on Banach spaces. New classes of generalized type-I functions are introduced for functions between Banach spaces. Based upon these generalized type-I functions, we obtain a few sufficient optimality conditions and prove some results on duality.

The gap function for optimization problems in Banach spaces

In this paper some properties of the gap function for multiple-objective optimization problems in Banach spaces are established.

Existence and porosity for a class of perturbed optimization problems in Banach spaces

Let X be a Banach space and Z a nonempty closed subset of X. Let J : Z → R be an upper semicontinuous function bounded from above. This paper is concerned with the perturbed optimization problem sup z ∈ Z { J ( z ) + ‖ x − z ‖ } , which is denoted by ( x , J ) -sup. We shall prove in the present paper that if Z is a closed boundedly relatively weakly compact nonempty subset, then the set of all x ∈ X for which the problem ( x , J ) -sup has a solution is a dense G δ -subset of X. In the case when X is uniformly convex and J is bounded, we will show that the set of all points x in X for which there does not exist z 0 ∈ Z such that J ( z 0 ) + ‖ x − z 0 ‖ = sup z ∈ Z { J ( z ) + ‖ x − z ‖ } is a σ-porous subset of X and the set of all points x ∈ X ∖ Z 0 such that there exists a maximizing sequence of the problem ( x , J ) -sup which has no convergent subsequence is a σ-porous subset of X ∖ Z 0 , where Z 0 denotes the set of all z ∈ Z such that z is in the solution set of ( z , J ) -sup.

Porosity of perturbed optimization problems in Banach spaces

Let X be a Banach space and Z a nonempty closed subset of X. Let J : Z → R be a lower semicontinuous function bounded from below. This paper is concerned with the perturbed optimization problem inf z ∈ Z { J ( z ) + ‖ x − z ‖ } , denoted by ( x , J ) -inf for x ∈ X . In the case when X is compactly fully 2-convex, it is proved in the present paper that the set of all points x in X for which there does not exist z 0 ∈ Z such that J ( z 0 ) + ‖ x − z 0 ‖ = inf z ∈ Z { J ( z ) + ‖ x − z ‖ } is a σ-porous set in X. Furthermore, if X is assumed additionally to be compactly locally uniformly convex, we verify that the set of all points x ∈ X ∖ Z 0 such that the problem ( x , J ) -inf fails to be approximately compact, is a σ-porous set in X ∖ Z 0 , where Z 0 denotes the set of all z ∈ Z such that z ∈ P Z ( z ) . Moreover, a counterexample to which some results of Ni [R.X. Ni, Generic solutions for some perturbed optimization problem in nonreflexive Banach space, J. Math. Anal. Appl. 302 (2005) 417–424] fail is provided.

Second‐Order Optimality Conditions for Scalar and Vector Optimization Problems in Banach Spaces

In this paper we present a very general and unified theory of second‐order optimality conditions for general optimization problems subject to abstract constraints in Banach spaces. Our results apply both to the scalar case and the multicriteria case. Our approach rests essentially on the use of a signed distance function for characterizing metric regularity of a certain multifunction associated with the problem. We prove variational results which show that, in a certain sense, our results are the best possible that one can obtain by using second‐order analysis. We demonstrate how recently devised optimality conditions can be derived from our general framework, how they can be extended under weakened assumptions from the scalar case to the multiobjective case, and even how some new results also can be obtained.

Descent methods for convex optimization problems in Banach spaces

We consider optimization problems in Banach spaces, whose cost functions are convex and smooth, but do not possess strengthened convexity properties. We propose a general class of iterative methods, which are based on combining descent and regularization approaches and provide strong convergence of iteration sequences to a solution of the initial problem.

Condition Number Theorems in Optimization

Condition numbers for optimization problems in Banach spaces are considered. Lower and upper estimates of the (suitably defined) distance from ill-conditioning are obtained in terms of the reciprocal of condition numbers. An approach is presented based on the metric regularity of the inverse to the arg min map. These results are extensions of the Eckart--Young distance theorem to optimization problems.

INEXACT VERSIONS OF PROXIMAL POINT AND AUGMENTED LAGRANGIAN ALGORITHMS IN BANACH SPACES

We generalize a proximal-like method for finding zeroes of maximal monotone operators in Hilbert spaces with quadratic regularization due to Solodov and Svaiter, making it possible the use of other kind of regularizations and extending it to Banach spaces. In particular, we introduce an appropriate error criterium to obtain an inexact proximal iteration based on Bregman functions and construct a hyperplane which strictly separates the current iterate from the solution set. A Bregman projection onto this hyperplane is then used to obtain the next iterate. Boundedness of the sequence and optimality of the weak accumulation points are established under suitable assumptions on the regularizing function, which hold for any power greater than one of the norm of any uniformly smooth and uniformly convex Banach space, without any assumption on the operator other than existence of zeroes. These assumptions let us, also, obtain similar results in Banach spaces for the Hybrid Extragradient-Generalized Proximal Point method, proposed by Solodov and Svaiter for finite dimensional spaces. We then transpose such methods to generate augmented Lagrangian methods for L P -constrained convex optimization problems in Banach spaces, obtaining two alternative procedures which allow for sequences, and optimality of primal and dual weak accumulation points, are then established, assuming only existence of Karush-Kuhn-Tucker pairs. *The work of this author was partially supported by CNPq grant no. 301280/86 and by Marcel Dassault chair, Centro de Modelamiento Matemático, Universidad de Chile.

On a proximal point method for convex optimization in banach spaces

We analyze the behavior of a parallel proximal point method for solving convex optimization problems in reflexive Banach spaces. Similar algorithms were known to converge under the implicit assumption that the norm of the space is Hilbertian. We extend the area of applicability of the proximal point method to solving convex optimization problems in Banach spaces on which totally convex functions can be found. This includes the class of all smooth uniformly convex Banach spaces. Also, our convergence results leave more flexibility for the choice of the penalty function involved in the algorithm and, in this way, allow simplification of the computational procedure.

A proximal point method for nonsmooth convex optimizationproblems in Banach spaces

In this paper we show the weak convergence and stability of the proximal point method when applied to the constrained convex optimization problem in uniformly convex and uniformly smooth Banach spaces. In addition, we establish a nonasymptotic estimate of convergence rate of the sequence of functional values for the unconstrained case. This estimate depends on a geometric characteristic of the dual Banach space, namely its modulus of convexity. We apply a new technique which includes Banach space geometry, estimates of duality mappings, nonstandard Lyapunov functionals and generalized projection operators in Banach spaces.

Nondifferentiable optimization in banach space

Necessary optimality conditions for nondifferentiable optimization problems in Banach spaces with its objective function and constraint function valued in a partial ordered vector space are given. The main results are obtained with the help of several Farkas type alternative theorems, one of them is proven in the paper for the first time only under a strict separation axiom of convex sets.

Perturbation analysis of optimization problems in banach spaces

In this paper we study local behavior of optimal solutions of parametric programs in Banach spaces. A general approach to approximation of the original problem by a simpler one is outlined. Under various regularity conditions, Lipschitz continuity and directional differentiability properties of the optimal solutions are derived.

Local stability of solutions to differentiable optimization problems in banach spaces

This paper considers a class of nonlinear differentiable optimization problems depending on a parameter. We show that, if constraint regularity, a second-order sufficient optimality condition, and a stability condition for the Lagrange multipliers hold, then for sufficiently smooth perturbations of the constraints and the objective function the optimal solutions locally obey a type of Lipschitz condition. The results are applied to finite-dimensional problems, equality constrained problems, and optimal control problems.

Second-Order Necessary Conditions for Nonlinear Optimization Problems in Banach Spaces and Their Application to an Optimal Control Problem

We study second-order necessary conditions for nonlinear optimization problems with equality and set constraints in Banach spaces. The necessary conditions are given in the primal form and the dual form by the use of second-order Neustadt derivative. Moreover we apply the necessary conditions to an optimal control problem without state constraints.

On necessary optimality conditions in vector optimization problems

Necessary conditions of the multiplier rule type for vector optimization problems in Banach spaces are proved by using separation theorems and Ljusternik's theorem. The Pontryagin maximum principle for multiobjective control problems with state constraints is derived from these general conditions. The paper extends to vector optimization results established in the scalar case by Ioffe and Tihomirov.

On the approximation of infinite optimization problems with an application to optimal control problems

This paper is concerned with approximations to infinite optimization problems in Banach spaces. Under the assumption of a first order necessary and a second order sufficient optimality condition we derive convergence results for the optimal solutions and the optimal values of the approximating problems. An application to finite difference approximations of nonlinear optimal control problems with state constraints is given.

Necessary Conditions in Nonsmooth Optimization

The paper contains four theorems concerning first order necessary conditions for a minimum in nonsmooth optimization problems in Banach spaces: a Lagrange multiplier rule for a mathematical programming problem in which an infinite dimensional equality constraint is included in the constraints, a general maximum principle for nonsmooth optimal control problems with state constraints, and a kind of multiplier rule for mathematical programming problems which applies when only finitely many equality constraints are present but when the Lipschitz continuity assumptions are removed. A summary of relevant background results from analysis is provided.