Higher-order Optimality Conditions Research Articles

Higher-order optimality conditions with separated derivatives and sensitivity analysis for set-valued optimization

In this paper, we establish optimality conditions and sensitivity analysis of set-valued optimization problems in terms of higher-order radial derivatives. First, we obtain the optimality conditions with separated derivatives for a set-valued optimization problem, here separated derivatives means the derivatives of objective and constraint functions are different. Then, some duality theorems for a mixed type of primal-dual set-valued optimization problem are gained. Finally, several results concerning higher-order sensitivity analysis are presented. The main results of this paper are illustrated by some concrete examples.

Higher-order KKT optimality conditions through contingent derivatives for constrained nonsmooth vector equilibrium problems

In this paper, we deal with some higher-order optimality conditions for local strict efficient solutions to a nonsmooth vector equilibrium problem with set, cone and equality constraints. For this aim, the concept of m−stable and m−steady functions (m≥2 and integer) for single-valued functions and some constraint qualifications of higher order in terms of contingent derivatives are proposed accordingly. We analyze the sum calculus rule of mth-order adjacent set, mth-order interior set, asymptotic mth-order tangent cone and asymptotic mth-order adjacent cone. Subsequently, we employ the obtained calculus rules to treat KKT necessary and sufficient optimality conditions of higher order in terms of contingent derivatives for the mth-order (local) strict efficient solutions to such problem. Simultaneously, we employ these rules to study KKT higher-order optimality conditions for such efficient solutions for a nonsmooth vector optimization problem with constraints. Some illustrative examples are provided to demonstrate the main results of the literature.

Higher-order optimality conditions of robust Benson proper efficient solutions in uncertain vector optimization problems

Higher-order optimality conditions of robust Benson proper efficient solutions in uncertain vector optimization problems

Higher-Order Optimality Conditions for Degenerate Unconstrained Optimization Problems

In this paper necessary and sufficient conditions of a minimum for the unconstrained degenerate optimization problem are presented. These conditions generalize the well-known optimality conditions. The new optimality conditions are presented in terms of polylinear forms and Hesse’s pseudoinverse matrix. The results are illustrated by examples.The formulation and appearance of these conditions differ from high-order optimality conditions by other authors. The suggested representation of high-order optimality conditions makes them convenient for the evaluation of the convergence rate for unconstrained optimization methods in the case of a singular minimum point, for example, for the analysis of Newton’s and quasi-Newton’s methods.

Higher-Order Optimality Conditions in Set-Valued Optimization with Respect to General Preference Mappings

We present higher order necessary conditions for a model of welfare economics, where the preference mapping has a star-shape property. We assume that the preferences can be satiable and can be described by an arbitrary preference set, without the use of utility functions. These conditions are formulated in terms of higher-order directional derivatives of multivalued mappings, and the variable domination structure is not given by cones.

Strict efficiency conditions for nonsmooth optimization with inclusion constraint under Hölder directional metric subregularity

In this paper, we investigate the higher-order optimality conditions for strict efficient solutions to a nonsmooth optimization problem subject to inclusion constraint. A concept of higher-order contingent derivative type for set-valued maps, some main calculus rules of which are obtained, is proposed and employed with the Robinson–Ursescu open mapping theorem to get a Karush–Kuhn–Tucker condition under assumptions of Hölder direction metric subregularity. Sufficient conditions for these solutions are established without convexity assumptions and possibly without the existence of derivatives. As an application, we extend some optimality conditions for a nonsmooth optimization problem subject to generalized inequality constraint. Another application is presented for necessary and sufficient conditions for robust local strict efficient solutions in uncertain vector optimization. Some examples are provided to illustrate our theorems as well. Our results are new and improve the existing ones in the literature substantially.

Studniarski's derivatives and efficiency conditions for constrained vector equilibrium problems with applications

This article is devoted to the investigation of a vector equilibrium problem involving equality, inequality and set constraints with nonsmooth functions via the higher-order Studniarski derivatives. Under the suitable constraint qualifications, higher-order Karush-Kuhn-Tucker type necessary efficiency conditions for the local weak efficient solutions of a constrained vector equilibrium problem are given. Under suitable assumptions on the objective and constraint functions, the higher-order Karush-Kuhn-Tucker type necessary optimality conditions for local weak efficient solutions become the sufficient conditions. As applications, we obtain the first-order and higher-order optimality conditions for the local weak efficient solutions of a constrained vector variational inequality problem, a constrained vector optimization problem and a transportation problem with two-sided constraints on supplies or demands in terms of Studniarski's derivatives. Several examples are also provided to illustrate the results of the paper.

Higher-order Optimality Conditions and Duality for Approximate Solutions in Non-convex Set-valued Optimization

This paper deals with higher-order optimality conditions and duality theory for approximate solutions in vector optimization involving non-convex set-valued maps. Firstly, under the assumption of near cone-subconvexlikeness for set-valued maps, the higher necessary and sufficient optimality conditions in terms of Studniarski derivatives are derived for local weak approximate minimizers of a set-valued optimization problem. Then, applications to Mond-Weir type dual problem are presented.

Second-Order Optimality and Beyond: Characterization and Evaluation Complexity in Convexly Constrained Nonlinear Optimization

High-order optimality conditions for convexly constrained nonlinear optimization problems are analysed. A corresponding (expensive) measure of criticality for arbitrary order is proposed and extended to define high-order epsilon -approximate critical points. This new measure is then used within a conceptual trust-region algorithm to show that if derivatives of the objective function up to order q ge 1 can be evaluated and are Lipschitz continuous, then this algorithm applied to the convexly constrained problem needs at most O(epsilon ^{-(q+1)}) evaluations of f and its derivatives to compute an epsilon -approximate qth-order critical point. This provides the first evaluation complexity result for critical points of arbitrary order in nonlinear optimization. An example is discussed, showing that the obtained evaluation complexity bounds are essentially sharp.

When the Karush–Kuhn–Tucker Theorem Fails: Constraint Qualifications and Higher-Order Optimality Conditions for Degenerate Optimization Problems

In this paper, we present higher-order analysis of necessary and sufficient optimality conditions for problems with inequality constraints. The paper addresses the case when the constraints are not assumed to be regular at a solution of the optimization problems. In the first two theorems derived in the paper, we show how Karush–Kuhn–Tucker necessary conditions reduce to a specific form containing the objective function only. Then we present optimality conditions of the Karush–Kuhn–Tucker type in Banach spaces under new regularity assumptions. After that, we analyze problems for which the Karush–Kuhn–Tucker form of optimality conditions does not hold and propose necessary and sufficient conditions for those problems. To formulate the optimality conditions, we introduce constraint qualifications for new classes of nonregular nonlinear optimization. The approach of p-regularity used in the paper can be applied to various degenerate nonlinear optimization problems due to its flexibility and generality.

Optimality conditions in set optimization employing higher-order radial derivatives

There are two approaches of defining the solutions of a set-valued optimization problem: vector criterion and set criterion. This note is devoted to higher-order optimality conditions using both criteria of solutions for a constrained set-valued optimization problem in terms of higher-order radial derivatives. In the case of vector criterion, some optimality conditions are derived for isolated (weak) minimizers. With set criterion, necessary and sufficient optimality conditions are established for minimal solutions relative to lower set-order relation.

Higher-Order Optimality Conditions and Higher-Order Tangent Sets

We present a simple approach to an analysis of higher order approximations to sets and functions. The objects we study are not of a specific order; they include objects of order 2 and $m$ with $m$ not necessarily an integer. We deduce from these concepts optimality conditions of higher order and we establish some calculus rules.

Higher order optimality conditions with an arbitrary non-differentiable function

In this paper, we introduce a higher order directional derivative and higher order subdifferential of Hadamard type of a given proper extended real function. We obtain necessary and sufficient optimality conditions of order n (n is a positive integer) for unconstrained problems in terms of them. We do not require any restrictions on the function in our results. In contrast to the most known directional derivatives, our derivative is harmonized with the classical higher order Fréchet directional derivative of the same order in the sense that both of them coincide, provided that the last one exists. A notion of a higher order critical direction is introduced. It is applied in the characterizations of the isolated local minimum of order n. Higher order invex functions are defined. They are the largest class such that the necessary conditions for a local minimum are sufficient for global one. We compare our results with some previous ones. As an application, we improve a result due to V. F. Demyanov, showing that the condition introduced by this author is a complete characterization of isolated local minimizers of order n.

Higher-order optimality conditions for strict and weak efficient solutions in set-valued optimization

In this paper, we introduce a notion of higher-order Studniarski epiderivative of a set-valued map and study its properties. Then, we discuss their applications to optimality conditions in set-valued optimization. Higher-order optimality conditions for strict and weak efficient solutions of a constrained set-valued optimization problem are established. Some remarks on the existing results in the literature are given from our results.

A note on “Higher-order optimality conditions in set-valued optimization using Studniarski derivatives and applications to duality” [Positivity. 18, 449–473(2014)

The note points out that the sufficiency of proposition 2.1 in Anh (Positivity 18:449–473, 2014) is erroneous and we provide an example to illustrate it. Also the proof of proposition 2.2 in Anh (Positivity 18:449–473, 2014) is incorrect and we give a new proof.

Higher-order optimality conditions for set-valued optimization with ordering cones having empty interior using variational sets

In this paper, we first establish chain rules and sum rules for variational sets of type 2. For their applications, optimality conditions of two particular optimization problems are discussed. Then, we obtain higher-order optimality conditions for proper Henig solutions of a set-valued optimization problem in terms of variational sets of type 2 when ordering cones have empty interior.

Necessary and sufficient conditions for a Pareto optimal allocation in a discontinuous Gale economic model

In this paper we examine the concept of Pareto optimality in a simplified Gale economic model without assuming continuity of the utility functions. We apply some existing results on higher-order optimality conditions to get necessary and sufficient conditions for a locally Pareto optimal allocation.

High-order optimality conditions for degenerate variational problems

The paper is devoted to the class of singular calculus of variations problems with constraints which are not regular mappings at the solution point. Methods of the p-regularity theory are used for investigation of isoperimetric and Lagrange singular problems. Necessary conditions for optimality in p-regular calculus of variations problem are presented.

Higher-order optimality conditions in set-valued optimization using Studniarski derivatives and applications to duality

In this paper, we introduce upper and lower Studniarski derivatives of set-valued maps. By virtue of these derivatives, higher-order necessary and sufficient optimality conditions are obtained for several kinds of minimizers of a set-valued optimization problem. Then, applications to duality are given. Some remarks on several existent results and examples are provided to illustrate our results.

Higher-order optimality conditions for proper efficiency in nonsmooth vector optimization using radial sets and radial derivatives

We establish both necessary and sufficient optimality conditions of higher orders for various kinds of proper solutions to nonsmooth vector optimization in terms of higher-order radial sets and radial derivatives. These conditions are for global solutions and do not require continuity and convexity assumptions. Examples are provided to show advantages of the results over existing ones in a number of cases.