Saddle points and gap functions for weak generalized Ky Fan inequalities

In this paper, we employ the image space analysis method to investigate a weak generalized Ky Fan inequality with cone constraints. Some regular weak separation functions are introduced, and generalized Lagrangian functions are constructed by using these regular weak separation functions. Under suitable convexity assumptions and Slater condition, the existence of solution for the weak generalized Ky Fan inequality with cone constraints is equivalent to a saddle point of the generalized Lagrangian functions. Moreover, we also use the regular weak separation functions to construct gap functions for the weak generalized Ky Fan inequality with cone constraints, and obtain its error bound.     

Nonlinear separation approach for the augmented Lagrangian in nonlinear semidefinite programming

This paper aims at showing that the class of augmented Lagrangian functions for nonlinear semidefinite programming problems can be derived, as a particular case, from a nonlinear separation scheme in the image space associated with the given problem. By means of the image space analysis, a global saddle point condition for the augmented Lagrangian function is investigated. It is shown that the existence of a saddle point is equivalent to a regular nonlinear separation of two suitable subsets of the image space. Without requiring the strict complementarity, it is proved that, under second order sufficiency conditions, the augmented Lagrangian function admits a local saddle point. The existence of global saddle points is then obtained under additional assumptions that do not require the compactness of the feasible set. Motivated by the result on global saddle points, we propose two modified primal-dual methods based on the augmented Lagrangian using different strategies and prove their convergence to a global solution and the optimal value of the original problem without requiring the boundedness condition of the multiplier sequence.     

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.     

Separation Approach for Augmented Lagrangians in Constrained Nonconvex Optimization

This paper aims at showing that the class of augmented Lagrangian functions, introduced by Rockafellar and Wets, can be derived, as a particular case, from a nonlinear separation scheme in the image space associated with the given problem; hence, it is part of a more general theory. By means of the image space analysis, local and global saddle-point conditions for the augmented Lagrangian function are investigated. It is shown that the existence of a saddle point is equivalent to a nonlinear separation of two suitable subsets of the image space. Under second-order sufficiency conditions in the image space, it is proved that the augmented Lagrangian admits a local saddle point. The existence of a global saddle point is then obtained under additional assumptions that do not require the compactness of the feasible set.     

Separation of sets and Wolfe duality

Lagrangian duality can be derived from separation in the Image Space, namely the space where the images of the objective and constraining functions of the given extremum problem run. By exploiting such a result, we analyse the relationships between Wolfe and Mond-Weir duality and prove their equivalence in the Image Space under suitable generalized convexity assumptions.      

Nonlinear Separation Approach to Constrained Extremum Problems

In this paper, by virtue of a nonlinear scalarization function, two nonlinear weak separation functions, a nonlinear regular weak separation function, and a nonlinear strong separation function are first introduced, respectively. Then, by the image space analysis, a global saddle-point condition for a nonlinear function is investigated. It is shown that the existence of a saddle point is equivalent to a nonlinear separation of two suitable subsets of the image space. Finally, some necessary and sufficient optimality conditions are obtained for constrained extremum problems.     

Some Results on Augmented Lagrangians in Constrained Global Optimization via Image Space Analysis

The aim of this paper is to present some results for the augmented Lagrangian function in the context of constrained global optimization by means of the image space analysis. It is first shown that a saddle point condition for the augmented Lagrangian function is equivalent to the existence of a regular nonlinear separation in the image space. Local and global sufficient optimality conditions for the exact augmented Lagrangian function are then investigated by means of second-order analysis in the image space. Local optimality result for this function is established under second-order sufficiency conditions in the image space. Global optimality result is further obtained under additional assumptions. Finally, it is proved that the exact augmented Lagrangian method converges to a global solution–Lagrange multiplier pair of the original problem under mild conditions.     

On Regularity for Constrained Extremum Problems. Part 2: Necessary Optimality Conditions

A particular theorem for linear separation between two sets is applied in the image space associated with a constrained extremum problem. In this space, the two sets are a convex cone, depending on the constraints (equalities and inequalities) of the given problem and the homogenization of its image. It is proved that the particular linear separation is equivalent to the existence of Lagrangian multipliers with a positive multiplier associated with the objective function (i.e., a necessary optimality condition). A comparison with the constraint qualifications and the regularity conditions existing in the literature is performed.     

Separation Functions and Optimality Conditions in Vector Optimization

In this paper, we propose weak separation functions in the image space for general constrained vector optimization problems on strong and weak vector minimum points. Gerstewitz function is applied to construct a special class of nonlinear separation functions as well as the corresponding generalized Lagrangian functions. By virtue of such nonlinear separation functions, we derive Lagrangian-type sufficient optimality conditions in a general context. Especially for nonconvex problems, we establish Lagrangian-type necessary optimality conditions under suitable restriction conditions, and we further deduce Karush–Kuhn–Tucker necessary conditions in terms of Clarke subdifferentials.     

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.     

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.     

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.     

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.     

Set-Valued Systems with Infinite-Dimensional Image and Applications

In infinite-dimensional spaces, we investigate a set-valued system from the image perspective. By exploiting the quasi-interior and the quasi-relative interior, we give some new equivalent characterizations of (proper, regular) linear separation and establish some new sufficient and necessary conditions for the impossibility of the system under new assumptions, which do not require the set to have nonempty interior. We also present under mild assumptions the equivalence between (proper, regular) linear separation and saddle points of Lagrangian functions for the system. These results are applied to obtain some new saddle point sufficient and necessary optimality conditions of vector optimization problems.     

Generalized -complementarity problems in Banach spaces

In this paper, a class of generalized f-complementarity problems and three classes of variational inequalities are introduced in real Banach spaces, and the equivalences among them are established under certain conditions. Several coercivity conditions are introduced for the existence of solutions of the generalized f-complementarity problem. Under some suitable assumptions, it is shown that each of these coercivity conditions is equivalent to the nonemptyness and boundedness of the solution set for the generalized f-complementarity problem in infinite-dimensional Banach spaces, and even the nonemptyness and compactness of the solution set for the generalized f-complementarity problem in finite-dimensional spaces. The existence of least elements for the feasible set of the generalized f-complementarity problem is also presented under suitable conditions.     

Gap functions for constrained vector variational inequalities with applications

In this paper, the image space analysis is applied to investigate scalar-valued gap functions and their applications for a (parametric)-constrained vector variational inequality. Firstly, using a non-linear regular weak separation function in image space, a gap function of a constrained vector variational inequality is obtained without any assumptions. Then, as an application of the gap function, two error bounds for the constrained vector variational inequality are derived by means of the gap function under some mild assumptions. Further, a parametric gap function of a parametric constrained vector variational inequality is presented. As an application of the parametric gap function, a sufficient condition for the continuity of the solution map of the parametric constrained vector variational inequality is established within the continuity and strict convexity of the parametric gap function. These assumptions do not include any information on the solution set of the parametric constrained vector variational inequality.     

Local saddle point and a class of convexification methods for nonconvex optimization problems

A class of general transformation methods are proposed to convert a nonconvex optimization problem to another equivalent problem. It is shown that under certain assumptions the existence of a local saddle point or local convexity of the Lagrangian function of the equivalent problem (EP) can be guaranteed. Numerical experiments are given to demonstrate the main results geometrically.     

Nonlinear separation approach to inverse variational inequalities

In this paper, we employ the image space analysis to investigate an inverse variational inequality (for short, IVI) with a cone constraint. By virtue of the nonlinear scalarization function commonly known as the Gerstewitz function, three nonlinear weak separation functions, two nonlinear regular weak separation functions and a nonlinear strong separation function are first introduced. Then, by these nonlinear separation functions, theorems of the weak and strong alternative and some optimality conditions for IVI with a cone constraint are derived without any convexity. In particular, a global saddle-point condition for a nonlinear function is investigated. It is shown that the existence of a saddle point is equivalent to a nonlinear separation of two suitable subsets of the image space. Finally, two gap functions and an error bound for IVI with a cone constraint are obtained.     

Local Convexification of the Lagrangian Function in Nonconvex Optimization

It is well-known that a basic requirement for the development of local duality theory in nonconvex optimization is the local convexity of the Lagrangian function. This paper shows how to locally convexify the Lagrangian function and thus expand the class of optimization problems to which dual methods can be applied. Specifically, we prove that, under mild assumptions, the Hessian of the Lagrangian in some transformed equivalent problem formulations becomes positive definite in a neighborhood of a local optimal point of the original problem.     

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.