On second-order optimality conditions for continuously Fréchet differentiable vector optimization problems

ABSTRACT

In this paper, we study a vector optimization problem (VOP) with both inequality and equality constraints. We suppose that the functions involved are Fréchet differentiable and their Fréchet derivatives are continuous or stable at the point of study. By virtue of a second-order constraint qualification of Abadie type, we provide second-order Karush–Kuhn–Tucker type necessary optimality conditions for the VOP. Moreover, we also obtain second-order sufficient optimality conditions for a kind of strict local efficiency. Both the necessary conditions and the sufficient conditions are shown in equivalent pairs of primal and dual formulations by using theorems of the alternative for the VOP.     

Second-Order Optimality Conditions in Set Optimization

In this paper, we propose second-order epiderivatives for set-valued maps. By using these concepts, second-order necessary optimality conditions and a sufficient optimality condition are given in set optimization. These conditions extend some known results in optimization.The authors are grateful to the referees for careful reading and helpful remarks.     

Optimal Control of the Instationary Three Dimensional Navier-Stokes-Voigt Equations

We study an optimal control problem with quadratic objective functional for the three dimensional Navier-Stokes-Voigt equations in bounded domains. We show the existence of optimal solutions, the necessary optimality conditions and the sufficient optimality conditions. The second-order optimality conditions obtained in the article seem to be optimal.     

Second-order invex functions in nonlinear programming

We introduce a notion of a second-order invex function. A Fréchet differentiable invex function without any further assumptions is second-order invex. It is shown that the inverse claim does not hold. A Fréchet differentiable function is second-order invex if and only if each second-order stationary point is a global minimizer. Two complete characterizations of these functions are derived. It is proved that a quasiconvex function is second-order invex if and only if it is second-order pseudoconvex. Further, we study the nonlinear programming problem with inequality constraints whose objective function is second-order invex. We introduce a notion of second-order type I objective and constraint functions. This class of problems strictly includes the type I invex ones. Then we extend a lot of sufficient optimality conditions with generalized convex functions to problems with second-order type I invex objective function and constraints. Additional optimality results, which concern type I and second-order type I invex data are obtained. An answer to the question when a kernel, which is not identically equal to zero, exists is given.     

Optimality Conditions for C1,1 Vector Optimization Problems

The main purpose of this paper is to make use of the second-order subdifferential of vector functions to establish necessary and sufficient optimality conditions for vector optimization problems.     

Optimality Conditions for C 1,1 Constrained Multiobjective Problems

The present paper is concerned with the study of the optimality conditions for constrained multiobjective programming problems in which the data have locally Lipschitz Jacobian maps. Second-order necessary and sufficient conditions for efficient solutions are established in terms of second-order subdifferentials of vector functions.     

First-and second-order optimality conditions for mathematical programs with vanishing constraints

We consider a special class of optimization problems that we call Mathematical Programs with Vanishing Constraints, MPVC for short, which serves as a unified framework for several applications in structural and topology optimization. Since an MPVC most often violates stronger standard constraint qualification, first-order necessary optimality conditions, weaker than the standard KKT-conditions, were recently investigated in depth. This paper enlarges the set of optimality criteria by stating first-order sufficient and second-order necessary and sufficient optimality conditions for MPVCs. Dedicated to Jiří V. Outrata on the occasion of his 60th birthday. This research was partially supported by the DFG (Deutsche Forschungsgemeinschaft) under grant KA1296/15-1.     

Note on local quadratic growth estimates in bang–bang optimal control problems

Abstract

Strong second-order conditions in mathematical programming play an important role not only as optimality tests but also as an intrinsic feature in stability and convergence theory of related numerical methods. Besides of appropriate firstorder regularity conditions, the crucial point consists in local growth estimation for the objective which yields inverse stability information on the solution. In optimal control, similar results are known in case of continuous control functions, and for bang–bang optimal controls when the state system is linear. The paper provides a generalization of the latter result to bang–bang optimal control problems for systems which are affine-linear w.r.t. the control but depend nonlinearly on the state. Local quadratic growth in terms of L₁ norms of the control variation are obtained under appropriate structural and second-order sufficient optimality conditions.     

No Gap Second-Order Optimality Conditions for Circular Conic Programs

In this article, we study the second-order optimality conditions for a class of circular conic optimization problem. First, the explicit expressions of the tangent cone and the second-order tangent set for a given circular cone are derived. Then, we establish the closed-form formulation of critical cone and calculate the “sigma” term of the aforementioned optimization problem. At last, in light of tools of variational analysis, we present the associated no gap second-order optimality conditions. Compared to analogous results in the literature, our approach is intuitive and straightforward, which can be manipulated and verified. An example is illustrated to this end.     

Second-Order Optimality Conditions in Generalized Semi-Infinite Programming

This paper deals with generalized semi-infinite optimization problems where the (infinite) index set of inequality constraints depends on the state variables and all involved functions are twice continuously differentiable. Necessary and sufficient second-order optimality conditions for such problems are derived under assumptions which imply that the corresponding optimal value function is second-order (parabolically) directionally differentiable and second-order epiregular at the considered point. These sufficient conditions are, in particular, equivalent to the second-order growth condition.     

Second-order optimality conditions for nondominated solutions of multiobjective programming with C 1,1 data

We examine new second-order necessary conditions and sufficient conditions which characterize nondominated solutions of a generalized constrained multiobjective programming problem. The vector-valued criterion function as well as constraint functions are supposed to be from the class C ^1,1. Second-order optimality conditions for local Pareto solutions are derived as a special case.     

Second-order conditions for extremum problems with nonregular equality constraints

Combining results of Avakov about tangent directions to equality constraints given by smooth operators with results of Ben-Tal and Zowe, we formulate a second-order theory for optimality in the sense of Dubovitskii-Milyutin which gives nontrivial conditions also in the case of equality constraints given by nonregular operators. Secondorder feasible and tangent directions are defined to construct conical approximations to inequality and equality constraints which within a single construction lead to first- and second-order conditions of optimality for the problem also in the nonregular case. The definitions of secondorder feasible and tangent directions given in this paper allow for reparametrizations of the approximating curves and give approximating sets which form cones. The main results of the paper are a theorem which states second-order necessary condition of optimality and several corollaries which treat special cases. In particular, the paper generalizes the Avakov result in the smooth case.This research was supported by NSF Grant DMS-91-009324, NSF Grant DMS-91-00043, SIUE Research Scholar Award and Fourth Quarter Fellowship, Summer 1992.     

Optimality conditions in a step control problem

First-and second-order necessary optimality conditions are obtained for the control of step systems.     

Second-order optimality conditions in set-valued optimization via asymptotic derivatives

In this article we give new second-order optimality conditions in set-valued optimization. We use the second-order asymptotic tangent cones to define second-order asymptotic derivatives and employ them to give the optimality conditions. We extend the well-known Dubovitskii–Milutin approach to set-valued optimization to express the optimality conditions given as an empty intersection of certain cones in the objective space. We also use some duality arguments to give new multiplier rules. By following the more commonly adopted direct approach, we also give optimality conditions in terms of a disjunction of certain cones in the image space. Several particular cases are discussed.     

Optimality Conditions and Duality in Nonsmooth Semiinfinite Programming

In this article, we utilize the semiinfinite versions of Guignard's constraint qualification and Motzkin's theorem of the alternative to establish a set of Karush–Kuhn–Tucker-type necessary optimality conditions for a nonsmooth and nonconvex semiinfinite programming problem. Furthermore, we discuss some sufficient optimality conditions and duality relations for our semiinfinite programming problem.     

Second-Order Optimality Conditions in Multiobjective Optimization Problems

In this paper, we develop second-order necessary and sufficient optimality conditions for multiobjective optimization problems with both equality and inequality constraints. First, we generalize the Lin fundamental theorem (Ref. 1) to second-order tangent sets; then, based on the above generalized theorem, we derive second-order necessary and sufficient conditions for efficiency.     

Modifications of the Wolfe Line Search Rules to Satisfy Second-Order Optimally Conditions in Unconstrained Optimization

The line search method incorporating the Wolfe conditions is modified to ensure that a descent algorithm terminates in a finite number of steps at an approximate stationary point where the second-order conditions of optimality are satisfied. A simple procedure based on conjugate directions is proposed to determine directions of negative curvature.     

Second-Order Optimality Conditions for Set-Valued Vector Equilibrium Problems

In this article, by using the generalized second-order contingent (adjacent) epiderivatives of set-valued maps, we obtain necessary optimality conditions and sufficient optimality conditions for weakly efficient solutions, Henig efficient solutions to the set-valued vector equilibrium problems with constraints. Some results of this article improve the corresponding results in literatures by lessening the assumption of convexity.     

Quartic Formulation of Standard Quadratic Optimization Problems

A standard quadratic optimization problem (StQP) consists of finding the largest or smallest value of a (possibly indefinite) quadratic form over the standard simplex which is the intersection of a hyperplane with the positive orthant. This NP-hard problem has several immediate real-world applications like the Maximum-Clique Problem, and it also occurs in a natural way as a subproblem in quadratic programming with linear constraints. To get rid of the (sign) constraints, we propose a quartic reformulation of StQPs, which is a special case (degree four) of a homogeneous program over the unit sphere. It turns out that while KKT points are not exactly corresponding to each other, there is a one-to-one correspondence between feasible points of the StQP satisfying second-order necessary optimality conditions, to the counterparts in the quartic homogeneous formulation. We supplement this study by showing how exact penalty approaches can be used for finding local solutions satisfying second-order necessary optimality conditions to the quartic problem: we show that the level sets of the penalty function are bounded for a finite value of the penalty parameter which can be fixed in advance, thus establishing exact equivalence of the constrained quartic problem with the unconstrained penalized version.     

The Legendre condition of the fractional calculus of variations

Fractional operators play an important role in modelling nonlocal phenomena and problems involving coarse-grained and fractal spaces. The fractional calculus of variations with functionals depending on derivatives and/or integrals of noninteger order is a rather recent subject that is currently in fast development due to its applications in physics and other sciences. In the last decade, several approaches to fractional variational calculus were proposed by using different notions of fractional derivatives and integrals. Although the literature of the fractional calculus of variations is already vast, much remains to be done in obtaining necessary and sufficient conditions for the optimization of fractional variational functionals, existence and regularity of solutions. Regarding necessary optimality conditions, all works available in the literature concern the derivation of first-order fractional conditions of Euler–Lagrange type. In this work, we obtain a Legendre second-order necessary optimality condition for weak extremizers of a variational functional that depends on fractional derivatives.