Optimal Potentials for Problems with Changing Sign Data

The main purpose of this paper is to study the duality and penalty method for a constrained nonconvex vector optimization problem. Following along with the image space analysis, a Lagrange-type duality for a constrained nonconvex vector optimization problem is proposed by virtue of the class of vector-valued regular weak separation functions in the image space. Simultaneously, some equivalent characterizations to the zero duality gap property are established including the Lagrange multiplier, the Lagrange saddle point and the regular separation. Moreover, an exact penalization is also obtained by means of a local image regularity condition and a class of particular regular weak separation functions in the image space.     

Unified Duality Theory for Constrained Extremum Problems. Part I: Image Space Analysis

This paper is concerned with a unified duality theory for a constrained extremum problem. Following along with the image space analysis, a unified duality scheme for a constrained extremum problem is proposed by virtue of the class of regular weak separation functions in the image space. Some equivalent characterizations of the zero duality property are obtained under an appropriate assumption. Moreover, some necessary and sufficient conditions for the zero duality property are also established in terms of the perturbation function. In the accompanying paper, the Lagrange-type duality, Wolfe duality and Mond–Weir duality will be discussed as special duality schemes in a unified interpretation. Simultaneously, three practical classes of regular weak separation functions will be also considered.     

Further Study on Augmented Lagrangian Duality Theory

In this paper, we present a necessary and sufficient condition for a zero duality gap between a primal optimization problem and its generalized augmented Lagrangian dual problems. The condition is mainly expressed in the form of the lower semicontinuity of a perturbation function at the origin. For a constrained optimization problem, a general equivalence is established for zero duality gap properties defined by a general nonlinear Lagrangian dual problem and a generalized augmented Lagrangian dual problem, respectively. For a constrained optimization problem with both equality and inequality constraints, we prove that first-order and second-order necessary optimality conditions of the augmented Lagrangian problems with a convex quadratic augmenting function converge to that of the original constrained program. For a mathematical program with only equality constraints, we show that the second-order necessary conditions of general augmented Lagrangian problems with a convex augmenting function converge to that of the original constrained program.This research is supported by the Research Grants Council of Hong Kong (PolyU B-Q359.)     

Coercivity and image of constrained extremum problems

The concept of coercivity is extended to the image space, and it is exploited to obtain a lower bound for the minimum of a constrained extremum problem and a sufficient condition for the optimality. Problems which are coercive in the image space turn out to have zero duality gap. A possible application to variational inequalities is illustrated by means of an example.     

On the Image Regularity Condition

In this paper, we continue the analysis of the image regularity condition (IRC) as introduced in a previous paper where we have proved that IRC implies the existence of generalized Lagrange-John multipliers with first component equal to 1. The term generalized is connected with the fact that the separation (in the image space) is not necessarily linear (when we have classic Lagrange-John multipliers), but it can be also nonlinear. Here, we prove that the IRC guarantees, also in the nondifferentiable case, the fact that 0 is a solution of the first-order homogeneized (linearized) problem obtained by means of the Dini-Hadamard derivatives.     

A generalized duality and applications

The aim of this paper is to present a nonconvex duality with a zero gap and its connection with convex duality. Since a convex program can be regarded as a particular case of convex maximization over a convex set, a nonconvex duality can be regarded as a generalization of convex duality. The generalized duality can be obtained on the basis of convex duality and minimax theorems. The duality with a zero gap can be extended to a more general nonconvex problems such as a quasiconvex maximization over a general nonconvex set or a general minimization over the complement of a convex set. Several applications are given.On leave from the Institute of Mathematics, Hanoi, Vietnam.     

Optimality conditions for vector optimization problems with non-cone constraints in image space

ABSTRACT

In this paper, we employ the image space analysis method to investigate a vector optimization problem with non-cone constraints. First, we use the linear and nonlinear separation techniques to establish Lagrange-type sufficient and necessary optimality conditions of the given problem under convexity assumptions and generalized Slater condition. Moreover, we give some characterizations of generalized Lagrange saddle points in image space without any convexity assumptions. Finally, we derive the vectorial penalization for the vector optimization problem with non-cone constraints by a general way.     

Gap Functions and Existence of Solutions to Generalized Vector Quasi-Equilibrium Problems

This paper deals with generalized vector quasi-equilibrium problems. By virtue of a nonlinear scalarization function, the gap functions for two classes of generalized vector quasi-equilibrium problems are obtained. Then, from an existence theorem for a generalized quasi-equilibrium problem and a minimax inequality, existence theorems for two classes of generalized vector quasi-equilibrium problems are established. This research is partially supported by the Postdoctoral Fellowship Scheme of The Hong Kong Polytechnic University and the National Natural Science Foundation of China.     

Some applications of the image space analysis to the duality theory for constrained extremum problems

By means of the Image Space Analysis, duality properties of a constrained extremum problem are investigated. The analysis of the lower semicontinuity of the perturbation function, related to a right-hand side perturbation of the given problem, leads to a characterization of zero duality gap in the image space.     

Vector quasi-equilibrium problems: separation,saddle points and error bounds for the solution set

In this paper, we employ the image space analysis (for short, ISA) to investigate vector quasi-equilibrium problems (for short, VQEPs) with a variable ordering relation, the constrained condition of which also consists of a variable ordering relation. The quasi relatively weak VQEP (for short, qr-weak VQEP) are defined by introducing the notion of the quasi relative interior. Linear separation for VQEP (res., qr-weak VQEP) is characterized by utilizing the quasi interior of a regularization of the image and the saddle points of generalized Lagrangian functions. Lagrangian type optimality conditions for VQEP (res., qr-weak VQEP) are then presented. Gap functions for VQEP (res., qr-weak VQEP) are also provided and moreover, it is shown that an error bound holds for the solution set of VQEP (res., qr-weak VQEP) with respect to the gap function under strong monotonicity.     

Solutions and optimality criteria for nonconvex constrained global optimization problems with connections between canonical and Lagrangian duality

This paper presents a canonical duality theory for solving a general nonconvex quadratic minimization problem with nonconvex constraints. By using the canonical dual transformation developed by the first author, the nonconvex primal problem can be converted into a canonical dual problem with zero duality gap. A general analytical solution form is obtained. Both global and local extrema of the nonconvex problem can be identified by the triality theory associated with the canonical duality theory. Illustrative applications to quadratic minimization with multiple quadratic constraints, box/integer constraints, and general nonconvex polynomial constraints are discussed, along with insightful connections to classical Lagrangian duality. Criteria for the existence and uniqueness of optimal solutions are presented. Several numerical examples are provided.     

A new approach to constrained optimization via image space analysis

In this article, by introducing a class of nonlinear separation functions, the image space analysis is employed to investigate a class of constrained optimization problems. Furthermore, the equivalence between the existence of nonlinear separation function and a saddle point condition for a generalized Lagrangian function associated with the given problem is proved.     

Stable zero duality gaps in convex programming: Complete dual characterisations with applications to semidefinite programs

The zero duality gap that underpins the duality theory is one of the central ingredients in optimisation. In convex programming, it means that the optimal values of a given convex program and its associated dual program are equal. It allows, in particular, the development of efficient numerical schemes. However, the zero duality gap property does not always hold even for finite-dimensional problems and it frequently fails for problems with non-polyhedral constraints such as the ones in semidefinite programming problems. Over the years, various criteria have been developed ensuring zero duality gaps for convex programming problems. In the present work, we take a broader view of the zero duality gap property by allowing it to hold for each choice of linear perturbation of the objective function of the given problem. Globalising the property in this way permits us to obtain complete geometric dual characterisations of a stable zero duality gap in terms of epigraphs and conjugate functions. For convex semidefinite programs, we establish necessary and sufficient dual conditions for stable zero duality gaps, as well as for a universal zero duality gap in the sense that the zero duality gap property holds for each choice of constraint right-hand side and convex objective function. Zero duality gap results for second-order cone programming problems are also given. Our approach makes use of elegant conjugate analysis and Fenchel's duality.     

集值映射的广义对称向量拟平衡问题

本文提出了一类集值映射的广义对称向量拟平衡问题．利用非线性标量化函数,分别在局部凸Hausdorff拓扑向量空间和一般的Hausdorff拓扑向量空间中,证明了广义对称向量拟平衡问题解的存在性定理．     

Nonlinear Lagrangian Theory for Nonconvex Optimization

The Lagrangian function in the conventional theory for solving constrained optimization problems is a linear combination of the cost and constraint functions. Typically, the optimality conditions based on linear Lagrangian theory are either necessary or sufficient, but not both unless the underlying cost and constraint functions are also convex.We propose a somewhat different approach for solving a nonconvex inequality constrained optimization problem based on a nonlinear Lagrangian function. This leads to optimality conditions which are both sufficient and necessary, without any convexity assumption. Subsequently, under appropriate assumptions, the optimality conditions derived from the new nonlinear Lagrangian approach are used to obtain an equivalent root-finding problem. By appropriately defining a dual optimization problem and an alternative dual problem, we show that zero duality gap will hold always regardless of convexity, contrary to the case of linear Lagrangian duality.     

The generalized Tykhonov well-posedness for system of vector quasi-equilibrium problems

In this paper, the notion of the generalized Tykhonov well-posedness for system of vector quasi-equilibrium problems are investigated. By using the gap functions of the system of vector quasi-equilibrium problems, we establish the equivalent relationship between the generalized Tykhonov well-posedness of the system of vector quasi-equilibrium problems and that of the minimization problems. We also present some metric characterizations for the generalized Tykhonov well-posedness of the system of vector quasi-equilibrium problems. The results in this paper are new and extend some known results in the literature.     

Nonlinear separation in the image space with applications to constrained optimization

In this paper, by means of the image space analysis, we obtain optimality conditions for vector optimization of objective multifunction with multivalued constraints based on disjunction of two suitable subsets of the image space. By the oriented distance function a nonlinear regular separation is introduced and some optimality conditions for the constrained extremum problem are obtained. It is shown that the existence of a nonlinear separation is equivalent to a saddle point condition for the generalized Lagrangian function.     

Nonlinear Lagrange Duality Theorems and Penalty Function Methods In Continuous Optimization

We propose a general dual program for a constrained optimization problem via generalized nonlinear Lagrangian functions. Our dual program includes a class of general dual programs with explicit structures as special cases. Duality theorems with the zero duality gap are proved under very general assumptions and several important corollaries which include some known results are given. Using dual functions as penalty functions, we also establish that a sequence of approximate optimal solutions of the penalty function converges to the optimal solution of the original optimization problem.     

Refinements on Gap Functions and Optimality Conditions for Vector Quasi-Equilibrium Problems via Image Space Analysis

By means of some new results on generalized systems, vector quasi-equilibrium problems with a variable ordering relation are investigated from the image perspective. Lagrangian-type optimality conditions and gap functions are obtained under mild generalized convexity assumptions on the given problem. Applications to the analysis of error bounds for the solution set of a vector quasi-equilibrium problem are also provided. These results are refinements of several authors’ works in recent years and also extend some corresponding results in the literature.     

The Nonnegative Zero-Norm Minimization Under Generalized Z-Matrix Measurement

In this paper, we consider the zero-norm minimization problem with linear equation and nonnegativity constraints. By introducing the concept of generalized Z-matrix for a rectangular matrix, we show that this zero-norm minimization with such a kind of measurement matrices and nonnegative observations can be exactly solved via the corresponding p-norm minimization with p in the open interval from zero to one. Moreover, the lower bound of sample number for exact recovery is allowed to be the same as the sparsity of the original image or signal by the underlying zero-norm minimization. A practical application in communications is presented, which satisfies the generalized Z-matrix recovery condition.