Condition Number Theorems in Linear-Quadratic Optimization

The condition number of a given mathematical problem is often related to the reciprocal of its distance from ill-conditioning. Such a property is proved here in the infinite-dimensional setting for linear-quadratic convex optimization of two types: linearly constrained convex quadratic problems, and minimum norm least squares solutions. A uniform version of such theorem is obtained in both cases for suitably equi-bounded classes of optimization problems. An application to the conditioning of a Ritz method is presented. For least squares problems it is shown that the semi-Fredholm property of the operators involved determines the validity of a condition number theorem.     

Technical efficiency and distance to a reverse convex set

The problem of finding the distance to a reverse (or complement of a) convex subset in a normed vector space is considered. This nonconvex and, in general, nonsmooth optimization problem arises in quantitative economics in the theory of measuring the technical efficiency of production units. In this context, applying a suitable duality theorem similar to the Nirenberg's one known for the distance to a convex subset, the problem reduces to a finite number of independent linear programming problems.     

On zero duality gap in nonconvex quadratic programming problems

We present in this paper new sufficient conditions for verifying zero duality gap in nonconvex quadratically/linearly constrained quadratic programs (QP). Based on saddle point condition and conic duality theorem, we first derive a sufficient condition for the zero duality gap between a quadratically constrained QP and its Lagrangian dual or SDP relaxation. We then use a distance measure to characterize the duality gap for nonconvex QP with linear constraints. We show that this distance can be computed via cell enumeration technique in discrete geometry. Finally, we revisit two sufficient optimality conditions in the literature for two classes of nonconvex QPs and show that these conditions actually imply zero duality gap.     

非凸集值映射的包含切性及应用

本文在一般的Ｂａｎａｃｈ空间Ｘ中研究从非空闭集ＫＸ到Ｘ的非凸集值映射Ｆ的包含切性问题．得到的结果定理３．１把有关的结论推广到非光滑空间，定理３．３则将有限维空间的正则性定理推广到任意的Ｂａｎａｃｈ空间．作为结果的应用，我们证明了无穷维非凸微分包含和非凸控制系统生存解的存在性，且给出了一个方便的等价切性条件．     

向量最优化弱有效解的特征和存在性

研究多目标凸向量优化问题在Gateaux可微条件下弱有效解的特性,并讨论一类非凸向量最优化问题弱有效解及与一变分不等式的等价性,给出了解的存在性。     

Nonconvex Separation Theorem for Multifunctions,Subdifferential Calculus and Applications

We derive a nonconvex separation theorem for multifunctions that generalizes an early result of Borwein and Jofré and show that this result is equivalent to several other subdifferential calculus results in smooth Banach spaces. Then we apply this nonconvex separation theorem to improve a second welfare theorem in economics and a necessary optimality condition for a multi-objective optimization problem.     

Nonconvex Differential Inclusions with Nonlinear Monotone Boundary Conditions

Existence results for problems with monotone nonlinear boundary conditions obtained in the previous publications by the author for functional differential equations are transferred to the case of nonconvex differential inclusions with the help of the selection theorem due to A. Bressan and G. Colombo.     

Generalized weak subdifferentials

In this article, generalized weak subgradient (gw-subgradient) and generalized weak subdifferential (gw-subdifferential) are defined for nonconvex functions with values in an ordered vector space. Convexity and closedness of the gw-subdifferential are stated and proved. By using the gw-subdifferential, it is shown that the epigraph of nonconvex functions can be supported by a cone instead of an affine subspace. A generalized lower (locally) Lipschitz function is also defined. By using this definition, some existence conditions of the gw-subdifferentiability of any function are stated and some properties of gw-subdifferentials of any function are examined. Finally, by using gw-subdifferential, a global minimality condition is obtained for nonconvex functions.     

Vector continuous-time programming without differentiability

In this work continuous-time programming problems of vector optimization are considered. Firstly, a nonconvex generalized Gordan’s transposition theorem is obtained. Then, the relationship with the associated weighting scalar problem is studied and saddle point optimality results are established. A scalar dual problem is introduced and duality theorems are given. No differentiability assumption is imposed.     

Lyapunov Sequences and a Turnpike Theorem without Convexity

A turnpike theorem is presented for a class of nonautonomous nonconvex difference inclusions defined by positively homogeneous increasing setvalued mappings. The proof involves a new concept of Lyapunov sequences based on Minkowski gauges of certain normal sets.     

Some Results on Condition Numbers in Convex Multiobjective Optimization

Various notions of condition numbers are used to study some sensitivity aspects of scalar optimization problems. The aim of this paper is to introduce a notion of condition number to study the case of a multiobjective optimization problem defined via m convex C ^1,1 objective functions on a given closed ball in ? ⁿ . Two approaches are proposed: the first one adopts a local point of view around a given solution point, whereas the second one considers the solution set as a whole. A comparison between the two notions of well-conditioned problem is developed. We underline that both the condition numbers introduced in the present work reduce to the same condition number proposed by Zolezzi in 2003, in the special case of the scalar optimization problem considered there. A pseudodistance between functions is defined such that the condition number provides an upper bound on how far from a well–conditioned function f a perturbed function g can be chosen in order that g is well–conditioned too. For both the local and the global approach an extension of classical Eckart–Young distance theorem is proved, even if only a special class of perturbations is considered.     

基于拟相对内部集值优化问题弱有效元的最优性条件

本文研究了基于拟相对内部的非凸集值优化问题弱有效元的最优性条件.首先,讨论了弱有效元与线性子空间之间的关系,利用涉及拟相对内部的凸集分离定理,获得了弱有效元的最优性条件.其次,给出了基于拟相对内部弱有效元的Lagrange乘子定理.     

关于局部凸空间中向量Ekeland变分原理的等价性

基于各种Ekeland变分原理的等价形式, 主要研究局部凸空间中给定有界凸子集乘以距离函数为扰动的单调半连续映射的向量Ekeand变分原理的等价性问题. 首先利用局部凸空间中的向量Ekeland变分原理证明了向量Caristi-Kirk不动点定理,向量 Takahashi非凸极小化定理和向量Oettli-Th\'{e}ra定理. 进一步研究了向量Ekeland变分原理与向量Caristi-Kirk不动点定理,向量Takahashi非凸极小化定理和向量Oettli-Th\'{e}ra定理的等价性.     

A note on the convergence of an algorithm for nonconvex programming problems

A strong convergence theorem is proven to hold for the general algorithm of the branch and bound type for solving nonconvex programming problems given in [1].     

Existence of Solutions for a Class of Nonlinear Evolution Inclusions with Nonlocal Conditions

This paper deals with the nonlocal problems for a class of nonlinear first-order evolution inclusions. Some existence results are established for the cases of a convex and of a nonconvex valued perturbation terms. Also, the existence of extremal solutions and a strong relaxation theorem are obtained. Subsequently a nonlinear hyperbolic optimal control problem is considered and the existence theorems based on the proven results are obtained. Then the nonlinear version of “bang–bang” principle for control systems is given as well by utilizing the relaxation theorem.     

Separation theorems for nonconvex sets and application in optimization

The aim of this paper is to present separation theorems for two disjoint closed sets, without convexity condition. First, a separation theorem for a given closed cone and a point outside from this cone, is proved and then it is used to prove a separation theorem for two disjoint sets. Illustrative examples are provided to highlight the important aspects of these theorems. An application to optimization is also presented to prove optimality condition for a nonconvex optimization problem.     

Boundary value problems of a class of nonlinear partial differential inclusions

This paper deals with the boundary value problems of nonlinear partial differential inclusions, driven by a negative Laplacian, and with the multivalued term which contains the gradient. It is proved the existence of solutions for the inclusions with the convex and nonconvex valued perturbations. The existence of extremal solutions and a strong relaxation theorem are also obtained.     

一类积分微分包含的周期解

研究了一类积分微分包含的周期解,利用Kakutani不动点定理和Tichonoff不动点定理给出了凸和非凸两种情形下周期解存在的充分条件.     

求解非凸半定规划的一个非线性Lagrange算法及其收敛性分析

本文提出了一个求解非凸半定规划的非线性Lagrange算法,当二阶充分条件以及严格互补条件成立时,证明了这一算法的收敛性定理.收敛结果表明,当惩罚参数小于某个阀值时,算法是局部收敛的;此外,还给出了解的一个依赖于惩罚参数的误差界.     

A Note on Nonconvex Minimax Theorem with Separable Homogeneous Polynomials

The minimax theorem for a convex-concave bifunction is a fundamental theorem in optimization and convex analysis, and has a lot of applications in economics. In the last two decades, a nonconvex extension of this minimax theorem has been well studied under various generalized convexity assumptions. In this note, by exploiting the hidden convexity (joint range convexity) of separable homogeneous polynomials, we establish a nonconvex minimax theorem involving separable homogeneous polynomials. Our result complements the existing study of nonconvex minimax theorem by obtaining easily verifiable conditions for the nonconvex minimax theorem to hold.