Second-order Constraint Qualifications Research Articles

Second-order optimality conditions for locally Lipschitz vector optimization problems

In this paper, we derive primal and dual second-order necessary and sufficient optimality conditions for a vector optimization problem with equality and inequality constraints where the functions involved are locally Lipschitz. We introduce a weaker notion of second-order Abadie constraint qualification to derive second-order necessary conditions for weak local Pareto minima and strict local Pareto minima of order two in terms of Páles and Zeidan's second-order upper directional derivatives. Dual necessary conditions are derived for both types of minimal solutions in finite-dimensional spaces assuming the functions to be first-order continuously Fréchet differentiable. In the same setting, we derive dual and primal second-order sufficient optimality conditions for strict local Pareto minima of order two in terms of second-order lower directional derivatives.

A Note on the McCormick Second-Order Constraint Qualification

The study of optimality conditions and constraint qualification is a key topic in nonlinear optimization. In this work, we present a reformulation of the well-known second-order constraint qualification described by McCormick in [17]. This reformulation is based on the use of feasible arcs, but is independent of Lagrange multipliers. Using such a reformulation, we can show that a local minimizer verifies the strong second-order necessary optimality condition. We can also prove that the reformulation is weaker than the known relaxed constant rank constraint qualification in [19]. Furthermore, we demonstrate that the condition is neither related to the MFCQ+WCR in [8] nor to the CCP2 condition, the companion constraint qualification associated with the second-order sequential optimality condition AKKT2 in [5].

Second-Order Optimality Conditions for Strict Pareto Minima and Weak Efficiency for Nonsmooth Constrained Vector Equilibrium Problems

In this article, some types of lower and upper second-order strictly pseudoconvexity are provided for establishing sufficient conditions for the second-order strict local Pareto minima of nonsmooth vector equilibrium problem with set, inequality and equality constraints. Based on the notion of Gerstewitz mappings, some Kuhn-Tucker-type multiplier rules for the strict local Pareto minima of such problem are obtained. We also construct the second-order constraint qualification in terms of first- and second-order directional derivatives of the (CQ) and (CQ1) types. Using this constraint qualifications, some second-order primal and dual necessary optimality conditions in terms of second-order upper and lower Dini directional derivatives for such minima are derived. Under suitable assumptions on the lower and upper strictly pseudoconvexity of order two of objective and constraint functions, second-order necessary optimality conditions become sufficient optimality conditions to such problem. Some illustrative examples are also given for our findings.

Second-order efficient optimality conditions for set-valued vector optimization in terms of asymptotic contingent epiderivatives

We propose a generalized second-order asymptotic contingent epiderivative of a set-valued mapping, study its properties, as well as relations to some second-order contingent epiderivatives, and sufficient conditions for its existence. Then, using these epiderivatives, we investigate set-valued optimization problems with generalized inequality constraints. Both second-order necessary conditions and sufficient conditions for optimality of the Karush–Kuhn–Tucker type are established under the second-order constraint qualification. An application to Mond–Weir and Wolfe duality schemes is also presented. Some remarks and examples are provided to illustrate our results.

Optimality conditions for circular cone complementarity programs

In this paper, we first provide some results on first- and second-order analysis to a circular cone complementary set Γ. In this way, we obtain a parabolical regularity of Γ as well as an exact computation for second epi-derivative of the indicator function to Γ. We then establish first- and second-order optimality conditions for a mathematical programming problem involving a circular cone complementary constraint under the validation of metric subregularity and second-order tangent constraint qualification. Our results are sharpened in comparison with some existing ones in the literature.

New Constraint Qualifications with Second-Order Properties in Nonlinear Optimization

In this paper, we present and discuss new constraint qualifications to ensure the validity of well-known second-order properties in nonlinear optimization. Here, we discuss conditions related to the so-called basic second-order condition, where a new notion of polar pairing is introduced in order to replace the polar operation, useful in the first-order case. We then proceed to define our second-order constraint qualifications, where we present an approach similar to the Guignard constraint qualification in the first-order case.

Notes on Constraint Qualifications for Second-Order Optimality Conditions

In the first part of this paper we point out some basic properties of the critical cones used in second-order optimality conditions and give a simple proof of a strong second-order necessary optimality condition by assuming a “modified” first-order Abadie constraint qualification. In the second part we give some insights on second-order constraint qualifications related to second-order local approximations of the feasible set.

Second-order optimality conditions and Lagrange multiplier characterizations of the solution set in quasiconvex programming

ABSTRACTSecond-order optimality conditions for the vector nonlinear programming problems with inequality constraints and continuously differentiable data are studied in this paper. We introduce a new second-order constraint qualification, which includes Mangasarian-Fromovitz constraint qualification as a particular case. We obtain necessary and sufficient conditions for weak efficiency of problems with a second-order pseudoconvex vector objective function and quasiconvex constraints. We also derive Lagrange multiplier characterizations of the solution set of a scalar problem with a second-order pseudoconvex objective function and quasiconvex inequality constraints, provided that one of the solutions and the Lagrange multipliers in the Karush-Kuhn-Tucker conditions are known. At last, we introduce notions of a second-order pseudoconvex vector function and KKT-pseudoconvex vector problem with inequality constraints. We derive necessary and sufficient conditions for efficiency in the vector problem with inequality constraints. Three examples are presented.

Locally Lipschitz vector optimization problems: second-order constraint qualifications, regularity condition and KKT necessary optimality conditions

In the present paper, we are concerned with a class of constrained vector optimization problems, where the objective functions and active constraint functions are locally Lipschitz at the referee point. Some second-order constraint qualifications of Zangwill type, Abadie type and Mangasarian–Fromovitz type as well as a regularity condition of Abadie type are proposed in a nonsmooth setting. The connections between these proposed conditions are established. They are applied to develop second-order Karush–Kuhn–Tucker necessary optimality conditions for local (weak, Geoffrion properly) efficient solutions to the considered problem. Examples are also given to illustrate the obtained results.

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

ABSTRACTIn 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.

Existence and Boundedness of Second-order Karush-Kuhn-Tucker Multipliers for Set-valued Optimization with Variable Ordering Structures

In this paper we investigate second-order Karush-Kuhn-Tucker multipliers for both local nondominated and local minimal points of set-valued optimization with variable ordering structures. We prove calculus rules of second-order contingent derivatives of index $\gamma \in \{0,1\}$ and use them to establish improved Karush-Kuhn-Tucker multiplier rules of nonclassical forms which involve separately such derivatives of the objective, constraint and ordering maps. The equivalence between the nonemptiness and boundedness of the multiplier sets in these rules and second-order constraint qualifications of the Kurcyusz-Robinson-Zowe and Mangasarian-Fromovitz types is demonstrated.

Second-order necessary efficiency conditions for nonsmooth vector equilibrium problems

This paper presents primal and dual second-order Fritz John necessary conditions for weak efficiency of nonsmooth vector equilibrium problems involving inequality, equality and set constraints in terms of the Pales–Zeidan second-order directional derivatives. Dual second-order Karush–Kuhn–Tucker necessary conditions for weak efficiency are established under suitable second-order constraint qualifications.

Second-Order Conditions for Open-Cone Minimizers and Firm Minimizers in Set-Valued Optimization Subject to Mixed Constraints

We consider second-order optimality conditions for set-valued optimization problems subject to mixed constraints. Such optimization models are useful in a wide range of practical applications. By using several kinds of derivatives, we obtain second-order necessary conditions for local Q-minimizers and local firm minimizers with attention to the envelope-like effect. Under the second-order Abadie constraint qualification, we get stronger necessary conditions. When the second-order Kurcyusz---Robinson---Zowe constraint qualification is imposed, our multiplier rules are of the Karush---Kuhn---Tucker type. Sufficient conditions for firm minimizers are established without any convexity assumptions. As an application, we extend and improve some recent existing results for nonsmooth mathematical programming.

On strong and weak second-order necessary optimality conditions for nonlinear programming

Second-order necessary optimality conditions play an important role in optimization theory. This is explained by the fact that most numerical optimization algorithms reduce to finding stationary points satisfying first-order necessary optimality conditions. As a rule, optimization problems, especially the high dimensional ones, have a lot of stationary points so one has to use second-order necessary optimality conditions to exclude nonoptimal points. These conditions are closely related to second-order constraint qualifications, which guarantee the validity of second-order necessary optimality conditions. In this paper, strong and weak second-order necessary optimality conditions are considered and their validity proved under so-called critical regularity condition at local minimizers.

Second-Order Optimality Conditions for Vector Problems with Continuously Fréchet Differentiable Data and Second-Order Constraint Qualifications

Some necessary and sufficient optimality conditions for inequality constrained problems with continuously differentiable data were obtained in the papers [I. Ginchev and V.I. Ivanov, Second-order optimality conditions for problems with C$\sp{1}$ data, J. Math. Anal. Appl., v. 340, 2008, pp. 646--657], [V.I. Ivanov, Optimality conditions for an isolated minimum of order two in C$\sp{1}$ constrained optimization, J. Math. Anal. Appl., v. 356, 2009, pp. 30--41] and [V. I. Ivanov, Second- and first-order optimality conditions in vector optimization, Internat. J. Inform. Technol. Decis. Making, 2014, DOI: 10.1142/S0219622014500540]. In the present paper, we continue these investigations. We obtain some necessary optimality conditions of Karush--Kuhn--Tucker type for scalar and vector problems. A new second-order constraint qualification of Zangwill type is introduced. It is applied in the optimality conditions.

On Second-Order Optimality Conditions for Vector Optimization

In this article, two second-order constraint qualifications for the vector optimization problem are introduced, that come from first-order constraint qualifications, originally devised for the scalar case. The first is based on the classical feasible arc constraint qualification, proposed by Kuhn and Tucker (Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, vol. 1, pp. 481–492, University of California Press, California, 1951) together with a slight modification of McCormick’s second-order constraint qualification. The second—the constant rank constraint qualification—was introduced by Janin (Math. Program. Stud. 21:110–126, 1984). They are used to establish two second-order necessary conditions for the vector optimization problem, with general nonlinear constraints, without any convexity assumption.

Constant-Rank Condition and Second-Order Constraint Qualification

The constant-rank condition for feasible points of nonlinear programming problems was defined by Janin (Math. Program. Study 21:127–138, 1984). In that paper, the author proved that the constant-rank condition is a first-order constraint qualification. In this work, we prove that the constant-rank condition is also a second-order constraint qualification. We define other second-order constraint qualifications.

Second-Order Conditions for Efficiency in Nonsmooth Multiobjective Optimization Problems

In this paper, we are concerned with a nonsmooth multiobjective optimization problem with inequality constraints. We introduce a second-order constraint qualification, which is a generalization of the Abadie constraint qualification and derive second-order Kuhn-Tucker type necessary conditions for efficiency under the constraint qualification. Moreover, we give some conditions which ensure the constraint qualification holds.

Second-order conditions and constraint qualifications in stability and sensitivity analysis of solutions to optimization problems in Hilbert spaces

The concepts of the strong second-order sufficient optimality condition and of the linear independence of gradients of active constraints play a crucial role in stability and sensitivity analysis of solutions to finite-dimensional mathematical programming problems. In this paper an attempt is made to use these concepts in stability and sensitivity analysis of solutions to cone-constrained optimization problems in Hilbert spaces. The abstract results are applied to optimal control problems for affine systems subject to state-space constraints.

Second-order necessary and sufficient conditions in multiobjective programming∗

In this paper we study second-order optimality conditions for the multi-objective programming problems with both inequality constraints and equality constraints. Two weak second-order constraint qualifications are introduced, and based on them we derive several second-order necessary conditions for a local weakly efficient solution. Two second-order sufficient conditions are also presented.