Stability Analysis for Parameterized Variational Systems with Implicit Constraints

In the paper we provide new conditions ensuring the isolated calmness property and the Aubin property of parameterized variational systems with constraints depending, apart from the parameter, also on the solution itself. Such systems include, e.g., quasi-variational inequalities and implicit complementarity problems. Concerning the Aubin property, possible restrictions imposed on the parameter are also admitted. Throughout the paper, tools from the directional limiting generalized differential calculus are employed enabling us to impose only rather weak (non- restrictive) qualification conditions. Despite the very general problem setting, the resulting conditions are workable as documented by some academic examples.

     

Variational analysis of circular cone programs

This paper conducts variational analysis of circular programs, which form a new class of optimization problems in nonsymmetric conic programming, important for optimization theory and its applications. First, we derive explicit formulas in terms of the initial problem data to calculate various generalized derivatives/co-derivatives of the projection operator associated with the circular cone. Then we apply generalized differentiation and other tools of variational analysis to establish complete characterizations of full and tilt stability of locally optimal solutions to parameterized circular programs.     

On the Coderivative of the Projection Operator onto the Second-order Cone

The limiting (Mordukhovich) coderivative of the metric projection onto the second-order cone $\mathbb{R}^{n}$ is computed. This result is used to obtain a sufficient condition for the Aubin property of the solution map of a parameterized second-order cone complementarity problem and to derive necessary optimality conditions for a mathematical program with a second-order cone complementarity problem among the constraints.     

Exceptional Family of Elements for a Variational Inequality Problem and its Applications

This paper introduces a new concept of exceptional family of elements (abbreviated, exceptional family) for a finite-dimensional nonlinear variational inequality problem. By using this new concept, we establish a general sufficient condition for the existence of a solution to the problem. Such a condition is used to develop several new existence theorems. Among other things, a sufficient and necessary condition for the solvability of pseudo-monotone variational inequality problem is proved. The notion of coercivity of a function and related classical existence theorems for variational inequality are also generalized. Finally, a solution condition for a class of nonlinear complementarity problems with so-called P _* -mappings is also obtained.     

A non-interior-point smoothing method for variational inequality problem

In this paper, we focus on the variational inequality problem. Based on the Fischer-Burmeister function with smoothing parameters, the variational inequality problem can be reformulated as a system of parameterized smooth equations, a non-interior-point smoothing method is presented for solving the problem. The proposed algorithm not only has no restriction on the initial point, but also has global convergence and local quadratic convergence, moreover, the local quadratic convergence is established without a strict complementarity condition. Preliminary numerical results show that the algorithm is promising.     

Existence theorems of solution to variational inequality problems

This paper introduces a new concept of exceptional family for variational inequality problems with a general convex constrained set. By using this new concept, the authors establish a general sufficient condition for the existence of a solution to the problem. This condition is weaker than many known solution conditions and it is also necessary for pseudomonotone variational inequalities. Suffi-cient solution conditions for a class of nonlinear complementarity problems with Po mappings are also obtained.     

Riemann流形上ρ-(η,d)-B不变凸的向量变分不等式及向量优化问题

该文研究了Riemann流形上的优化问题.首先,利用广义方向导数在Riemann流形上引入ρ-(η,d)-B不变凸函数、ρ-(η,d)-B伪不变凸函数和ρ-(η,d)-B拟不变凸函数.其次,讨论了变分不等式的解与Riemann流形上向量优化问题解之间的关系.最后,建立了优化问题的Kuhn-Tucker充分条件.     

On the Directional Differentiability of the Solution Mapping for a Class of Variational Inequalities of the Second Kind

The directional differentiability of the solution mapping for a class of variational inequalities of the second kind inspired by applications in fluid mechanics and moving free boundary problems is investigated. The result is particularly relevant for the model predictive control or optimal control of such variational inequalities in that it can be used to derive stationarity conditions and efficient numerical methods.     

Implicit multifunction theorems for the sensitivity analysis of variational conditions

We study implicit multifunctions (set-valued mappings) obtained from inclusions of the form 0∈M(p,x), whereM is a multifunction. Our basic implicit multifunction theorem provides an approximation for a generalized derivative of the implicit multifunction in terms of the derivative of the multifunctionM. Our primary focus is on three special cases of inclusions 0∈M(p,x) which represent different kinds of generalized variational inequalities, called “variational conditions”. Appropriate versions of our basic implicit multifunction theorem yield approximations for generalized derivatives of the solutions to each kind of variational condition. We characterize a well-known generalized Lipschitz property in terms of generalized derivatives, and use our implicit multifunction theorems to state sufficient conditions (and necessary in one case) for solutions of variational conditions to possess this Lipschitz, property. We apply our results to a general parameterized nonlinear programming problem, and derive a new second-order condition which guarantees that the stationary points associated with the Karush-Kuhn-Tucker conditions exhibit generalized Lipschitz continuity with respect to the parameter.     

Optimality conditions for vector optimization problem governed by the cone constrained generalized equations

ABSTRACT

The paper considers a class of vector optimization problems with cone constrained generalized equations. By virtue of advanced tools of variational analysis and generalized differentiation, a limiting normal cone of the graph of the normal cone constrained by the second-order cone is obtained. Based on the calmness condition, we derive an upper estimate of the coderivative for a composite set-valued mapping. The necessary optimality condition for the problem is established under the linear independent constraint qualification. As a special case, the obtained optimality condition is simplified with the help of strict complementarity relaxation conditions. The numerical results on different examples are given to illustrate the efficiency of the optimality conditions.     

New extragradient-type methods for general variational inequalities

In this paper, we consider and analyze a new class of extragradient-type methods for solving general variational inequalities. The modified methods converge for pseudomonotone operators which is weaker condition than monotonicity. Our proof of convergence is very simple as compared with other methods. The proposed methods include several new and known methods as special cases. Our results present a significant improvement of previously known methods for solving variational inequalities and related optimization problems.     

On proto-differentiability of generalized perturbation maps

This paper is devoted to the sensitivity analysis in optimization problems and variational inequalities. The concept of proto-differentiability of set-valued maps (see [R.T. Rockafellar, Proto-differentiability of set-valued mappings and its applications in optimization, Ann. Inst. H. Poincaré Anal. Non Linéaire 6 (1989) 449-482]) plays the key role in our investigation. It is proved that, under some suitable qualification conditions, the generalized perturbation maps (that is, the solution set map to a parameterized constraint system, to a parameterized variational inequality, or to a parameterized optimization problem) are proto-differentiable.     

A Minty variational principle for set optimization

Extremal problems are studied involving an objective function with values in (order) complete lattices of sets generated by so-called set relations. Contrary to the popular paradigm in vector optimization, the solution concept for such problems, introduced by F. Heyde and A. Löhne, comprises the attainment of the infimum as well as a minimality property. The main result is a Minty type variational inequality for set optimization problems which provides a sufficient optimality condition under lower semicontinuity assumptions and a necessary condition under appropriate generalized convexity assumptions. The variational inequality is based on a new Dini directional derivative for set-valued functions which is defined in terms of a “lattice difference quotient.” A residual operation in a lattice of sets replaces the inverse addition in linear spaces. Relationships to families of scalar problems are pointed out and used for proofs. The appearance of improper scalarizations poses a major difficulty which is dealt with by extending known scalar results such as Diewert's theorem to improper functions.     

Extragradient methods for solving nonconvex variational inequalities

In this paper, we introduce and consider a new class of variational inequalities, which are called the nonconvex variational inequalities. Using the projection technique, we suggest and analyze an extragradient method for solving the nonconvex variational inequalities. We show that the extragradient method is equivalent to an implicit iterative method, the convergence of which requires only pseudo-monotonicity, a weaker condition than monotonicity. This clearly improves on the previously known result. Our method of proof is very simple as compared with other techniques.     

Lipschitzian stability of parametric variational inequalities over generalized polyhedra in Banach spaces

This paper concerns the study of solution maps to parameterized variational inequalities over generalized polyhedra in reflexive Banach spaces. It has been recognized that generalized polyhedral sets are significantly different from the usual convex polyhedra in infinite dimensions and play an important role in various applications to optimization, particularly to generalized linear programming. Our main goal is to fully characterize robust Lipschitzian stability of the aforementioned solution maps entirely via their initial data. This is done on the basis of the coderivative criterion in variational analysis via efficient calculations of the coderivative and related objects for the systems under consideration. The case of generalized polyhedra is essentially more involved in comparison with usual convex polyhedral sets and requires developing elaborated techniques and new proofs of variational analysis.     

On Lipschitz Semicontinuity Properties of Variational Systems with Application to Parametric Optimization

In this paper, two properties of recognized interest in variational analysis, known as Lipschitz lower semicontinuity and calmness, are studied with reference to a general class of variational systems, i.e. to solution mappings to parameterized generalized equations. In the consideration of the metric nature of such properties, some related sufficient conditions are established, which are expressed via nondegeneracy conditions on derivative-like objects appropriate for a metric space analysis. For certain classes of generalized equations in Asplund spaces, it is shown how such conditions can be formulated by using the Fréchet coderivative of the field and the derivative of the base. Applications to the stability analysis of parametric constrained optimization problems are proposed.     

Linearly perturbed generalized polyhedral normal cone mappings and applications

Under a mild regularity assumption, we derive an exact formula for the Fréchet coderivative and some estimates for the Mordukhovich coderivative of the normal cone mappings of perturbed generalized polyhedra in reflexive Banach spaces. Assume in addition that the generating elements are linearly independent and some qualification condition holds, the Lipschitzian stability of the parameterized variational inequalities over the right-hand side perturbed generalized polyhedra is characterized using the initial data.     

Necessary condition for eigensolutions of the 1-Laplace operator by means of inner variations

Eigenfunctions of the $p$ -Laplace operator for $p>1$ are defined to be critical points of an associated variational problem or, equivalently, to be solutions of the corresponding Euler–Lagrange equation. In the highly degenerated limit case of the 1-Laplace operator eigenfunctions can also be defined to be critical points of the corresponding variational problem if critical points are understood on the basis of the weak slope. However, the associated Euler–Lagrange equation has many solutions that are not critical points and, thus, it cannot be used for an equivalent definition. The present paper provides a new necessary condition for eigenfunctions of the 1-Laplace operator by means of inner variations of the associated variational problem and it is shown that this condition rules out certain solutions of the Euler–Lagrange equation that are not eigenfunctions.