Second-order variational analysis and characterizations of tilt-stable optimal solutions in infinite-dimensional spaces

The paper is devoted to developing second-order tools of variational analysis and their applications to characterizing tilt-stable local minimizers of constrained optimization problems infinite-dimensional spaces with many results new also in finite-dimensional settings. The importance of tilt stability has been well recognized from both theoretical and numerical aspects of optimization. Based on second-order generalized differentiation, we obtain qualitative and quantitative characterizations of tilt stability in general frameworks of constrained optimization and establish its relationships with strong metric regularity of subgradient mappings and uniform second-order growth. The results obtained are applied to deriving new necessary and sufficient conditions for tilt-stable minimizers in problems of nonlinear programming with twice continuously differentiable data in Hilbert spaces.     

Generalized Euler identity for subdifferentials of homogeneous functions and applications

In this paper, we mainly consider subdifferentials and basic subdifferentials of homogeneous functions defined on real Banach space and Asplund space respectively, and obtain the generalized Euler identity. As applications, we consider constrained optimization problems and several geometric properties of Banach space.     

Stability of Discrete Approximations and Necessary Optimality Conditions for Delay-Differential Inclusions

This paper is devoted to the study of optimization problems for dynamical systems governed by constrained delay-differential inclusions with generally nonsmooth and nonconvex data. We provide a variational analysis of the dynamic optimization problems based on their data perturbations that involve finite-difference approximations of time-derivatives matched with the corresponding perturbations of endpoint constraints. The key issue of such an analysis is the justification of an appropriate strong stability of optimal solutions under finite-dimensional discrete approximations. We establish the required pointwise convergence of optimal solutions and obtain necessary optimality conditions for delay-differential inclusions in intrinsic Euler–Lagrange and Hamiltonian forms involving nonconvex-valued subdifferentials and coderivatives of the initial data.     

Approximate subdifferential and metric regularity: The finite-dimensional case

The notion of graphical metric regularity is introduced and conditions ensuring this kind of regularity for systems of finite-dimensional multifunctions are given in terms of partial approximate subdifferentials.     

Expressions for subdifferential and optimization problems defined by nonsmooth homogeneous functions

In this paper, we prove a theoretical expression for subdifferentials of lower semicontinuous and homogeneous functions. The theoretical expression is a generalization of the Euler formula for differentiable homogeneous functions. As applications of the generalized Euler formula, we consider constrained optimization problems defined by nonsmooth positively homogeneous functions in smooth Banach spaces. Some results concerning Karush–Kuhn–Tucker points and necessary optimality conditions for the optimization problems are obtained.     

Second-Order Optimality Conditions for Strict Efficiency of Constrained Set-Valued Optimization

In this paper, we propose several second-order derivatives for set-valued maps and discuss their properties. By using these derivatives, we obtain second-order necessary optimality conditions for strict efficiency of a set-valued optimization problem with inclusion constraints in real normed spaces. We also establish second-order sufficient optimality conditions for strict efficiency of the set-valued optimization problem in finite-dimensional normed spaces. As applications, we investigate second-order sufficient and necessary optimality conditions for a strict local efficient solution of order two of a nonsmooth vector optimization problem with an abstract set and a functional constraint.     

Optimality Conditions for C 1,1 Constrained Multiobjective Problems

The present paper is concerned with the study of the optimality conditions for constrained multiobjective programming problems in which the data have locally Lipschitz Jacobian maps. Second-order necessary and sufficient conditions for efficient solutions are established in terms of second-order subdifferentials of vector functions.     

Local stability of solutions to differentiable optimization problems in banach spaces

This paper considers a class of nonlinear differentiable optimization problems depending on a parameter. We show that, if constraint regularity, a second-order sufficient optimality condition, and a stability condition for the Lagrange multipliers hold, then for sufficiently smooth perturbations of the constraints and the objective function the optimal solutions locally obey a type of Lipschitz condition. The results are applied to finite-dimensional problems, equality constrained problems, and optimal control problems.     

Lagrange necessary conditions for Pareto minimizers in Asplund spaces and applications

In this paper, new necessary conditions for Pareto minimal points to sets and Pareto minimizers for constrained multiobjective optimization problems are established without the sequentially normal compactness property and the asymptotical compactness condition imposed on closed and convex ordering cones in Bao and Mordukhovich [10] and Durea and Dutta [5], respectively. Our approach is based on a version of the separation theorem for nonconvex sets and the subdifferentials of vector-valued and set-valued mappings. Furthermore, applications in mathematical finance and approximation theory are discussed.     

Weak sharp efficiency in multiobjective optimization

By using the generalized Fermat rule, the Mordukhovich subdifferential for maximum functions, the fuzzy sum rule for Fréchet subdifferentials and the sum rule for Mordukhovich subdifferentials, we establish a necessary optimality condition for the local weak sharp efficient solution of a constrained multiobjective optimization problem. Moreover, by employing the approximate projection theorem, and some appropriate convexity and affineness conditions, we also obtain some sufficient optimality conditions respectively for the local and global weak sharp efficient solutions of such a multiobjective optimization problem.     

Discrete Approximations to Continuum Optimal Flow Problems

Problems in partial differential equations with inequality constraints can be used to describe a continuum analog to various optimal flow/cut problems. While general concepts from convex optimization (like duality) carry over into continuum problems, the application of ideas and algorithms from linear programming and network flow problems is challenging. The capacity constraints are nonlinear (but convex).
In this article, we investigate a discretized version of the planar maximum flow problem that preserves the nonlinear capacity constraints of the continuum problem. The resulting finite-dimensional problem can be cast as a second-order cone programming problem or a quadratically constrained program. Good numerical results can be obtained using commercial solvers. These results are in agreement with the continuum theory of a "challenge" problem posed by Strang.     

Conjugate Duality for Constrained Vector Optimization in Abstract Convex Frame

This article focuses on a conjugate duality for a constrained vector optimization in the framework of abstract convexity. With the aid of the extension for the notion of infimum to the vector space, a set-valued topical function and the corresponding conjugate map, subdifferentials are presented. Following this, a conjugate dual problem is proposed via this conjugate map. Then, inspired by some ideas in the image space analysis, some equivalent characterizations of the zero duality gap are established by virtue of the subdifferentials.     

Metric Inequality, Subdifferential Calculus and Applications

In this paper we establish characterizations of Asplund spaces in terms of conditions ensuring the metric inequality and intersection formulae. Then we establish chain rules for the limiting Fréchet subdifferentials. Necessary conditions for constrained optimization problems with non-Lipschitz data are derived.     

Second-order subdifferentials of C 1,1 functions and optimality conditions

We present second-order subdifferentials of Clarke's type of C ^1,1 functions, defined in Banach spaces with separable duals. One of them is an extension of the generalized Hessian matrix of such functions in ? ⁿ , considered by J. B. H.-Urruty, J. J. Strodiot and V. H. Nguyen. Various properties of these subdifferentials are proved. Second-order optimality conditions (necessary, sufficient) for constrained minimization problems with C ^1,1 data are obtained.     

Directional subdifferentials and optimality conditions

This paper is devoted to the introduction and development of new dual-space constructions of generalized differentiation in variational analysis, which combine certain features of subdifferentials for nonsmooth functions (resp. normal cones to sets) and directional derivatives (resp. tangents). We derive some basic properties of these constructions and apply them to optimality conditions in problems of unconstrained and constrained optimization.     

Limiting behavior of the approximate second-order subdifferential of a convex function

Hiriart-Urruty and the author recently introduced the notions of Dupin indicatrices for nonsmooth convex surfaces and studied them in connection with their concept of a second-order subdifferential for convex functions. They noticed that second-order subdifferentials can be viewed as limit sets of difference quotients involving approximate subdifferentials. In this paper, we elaborate this point in a more detailed way and discuss some related questions.The author is grateful to the referees for their helpful comments.     

集值映射的次微分以及最优性条件

基于已有的集值映射的弱次微分的概念,定义了集值映射的Henig全局次微分,研究了它的存在性条件以及运算性质．利用这一概念,分别给出了具约束向量集值最优化问题的Henig全局有效解对的必要性条件和充分性条件．     

Second-order subdifferentials and convexity of real-valued functions

In this paper we show that the positive semi-definiteness (PSD) of the Fréchet and/or Mordukhovich second-order subdifferentials can recognize the convexity of C¹ functions. However, the PSD is insufficient for ensuring the convexity of a locally Lipschitz function in general. A complete characterization of strong convexity via the second-order subdifferentials is also given.     

Error Bounds and Implicit Multifunction Theorem in Smooth Banach Spaces and Applications to Optimization

In this paper we provide an error bound estimate and an implicit multifunction theorem in terms of smooth subdifferentials and abstract subdifferentials. Then, we derive a subdifferential calculus and Fritz–John type necessary optimality conditions for constrained minimization problems.     

Partially Strictly Monotone and Nonlinear Penalty Functions for Constrained Mathematical Programs

We introduce the concept of partially strictly monotone functions and apply it to construct a class of nonlinear penalty functions for a constrained optimization problem. This class of nonlinear penalty functions includes some (nonlinear) penalty functions currently used in the literature as special cases. Assuming that the perturbation function is lower semi-continuous, we prove that the sequence of optimal values of nonlinear penalty problems converges to that of the original constrained optimization problem. First-order and second-order necessary optimality conditions of nonlinear penalty problems are derived by converting the optimality of penalty problems into that of a smooth constrained vector optimization problem. This approach allows for a concise derivation of optimality conditions of nonlinear penalty problems. Finally, we prove that each limit point of the second-order stationary points of the nonlinear penalty problems is a second-order stationary point of the original constrained optimization problem.