Global weak sharp minima for convex (semi-)infinite optimization problems

We mainly consider global weak sharp minima for convex infinite and semi-infinite optimization problems (CIP). In terms of the normal cone, subdifferential and directional derivative, we provide several characterizations for (CIP) to have global weak sharp minimum property.     

Generalized weak sharp minima in cone-constrained convex optimization with applications

In this paper, we consider convex optimization problems with cone constraints (CPC in short). We study generalized weak sharp minima properties for (CPC) in the Banach space and Hilbert space settings, respectively. Some criteria and characterizations for the solution set to be a set of generalized weak sharp minima for (CPC) are derived. As an application, we propose an algorithm for (CPC) in the Hilbert space setting. Convergence analysis of this algorithm is given.     

Necessary optimality conditions for weak sharp minima in set-valued optimization

The aim of the present paper is to get necessary optimality conditions for a general kind of sharp efficiency for set-valued mappings in infinite dimensional framework. The efficiency is taken with respect to a closed convex cone and as the basis of our conditions we use the Mordukhovich generalized differentiation. We have divided our work into two main parts concerning, on the one hand, the case of a solid ordering cone and, on the other hand, the general case without additional assumptions on the cone. In both situations, we derive some scalarization procedures in order to get the main results in terms of the Mordukhovich coderivative, but in the general case we also carryout a reduction of the sharp efficiency to the classical Pareto efficiency which, in addition with a new calculus rule for Fréchet coderivative of a difference between two maps, allows us to obtain some results in Fréchet form.     

Optimality conditions for weak {\psi} -sharp minima in vector optimization problems

In this paper, we develop the characterization of weak ${\psi}$ -sharp minimizer by means of an oriented distance function and investigate the weak ${\psi}$ -sharp minimizer of the composition of two functions. Moreover, we establish sufficient conditions of the weak ${\psi_1}$ -sharp and ${\psi_2}$ -sharp Pareto minimality for vector optimization problem with strictly differentiable and twice strictly differentiable objective function, respectively. Our results extend the corresponding ones in the literature.     

Generalized weak sharp minima in cone-constrained convex optimization on Hadamard manifolds

In this paper, a convex optimization problem with cone constraint (for short, CPC) is introduced and studied on Hadamard manifolds. Some criteria and characterizations for the solution set to be a set of generalized global weak sharp minima, generalized local weak sharp minima and generalized bounded weak sharp minima for (CPC) are derived on Hadamard manifolds.     

New characterizations of weak sharp minima

In this paper, we study the concept of weak sharp minima by using conjugate functions. Not only some well-known results can be obtained in this unified way, but also new characterizations are developed. Finally, under rather weak conditions, we establish the finite termination property for convex programming and variational inequality problem, respectively.     

Complete characterizations of local weak sharp minima with applications to semi-infinite optimization and complementarity

In this paper, we identify a favorable class of nonsmooth functions for which local weak sharp minima can be completely characterized in terms of normal cones and subdifferentials, or tangent cones and subderivatives, or their mixture in finite-dimensional spaces. The results obtained not only extend previous ones in the literature, but also allow us to provide new types of criteria for local weak sharpness. Applications of the developed theory are given to semi-infinite programming and to a new class of semi-infinite complementarity problems.     

On upper semicontinuity of global minima in constrained optimization problems

This paper studies stability properties of solutions for optimization problems subject to perturbations in constraints. For problems formulated in a complete metric space sufficient conditions for topological upper semicontinuity of the solution multifunction are derived without any compactness assumptions. When the feasible solution set is additionally assumed to be compact these conditions reduce to that of Berge (“Topological Spaces,” Macmillan Co., New York, 1963), and in that case they turn out to be also necessary.     

Characterizations of strict local minima and necessary conditions for weak sharp minima

In nonlinear programming, sufficient conditions of orderm usually identify a special type of local minimizer, here termed a strict local minimizer of orderm. In this paper, it is demonstrated that, if a constraint qualification is satisfied, standard sufficient conditions often characterize this special sort of minimizer. The first- and second-order cases are treated in detail. Necessary conditions for weak sharp local minima of orderm, a larger class of local minima, are also presented.This paper was completed during a sabbatical leave at the University of Waterloo. The author is grateful for the help and support of the Department of Combinatorics and Optimization. The helpful comments of the referees are also appreciated.     

A note on weak sharp minima in multicriteria linear programming

In this note, the set of weak Pareto solutions of a multicriteria linear programming problem (MCLP, for short) is proved to be a set of weak sharp minima for another residual function of MCLP, i.e., the minimum of the natural residual functions of finitely many scalarization problems of MCLP, which is less than the natural residual function of MCLP. This can be viewed as a slight improvement of a result due to Deng and Yang. Some examples are given to illustrate these results.     

Univariate parameterization for global optimization of mixed-integer polynomial problems

This paper presents a new relaxation technique to globally optimize mixed-integer polynomial programming problems that arise in many engineering and management contexts. Using a bilinear term as the basic building block, the underlying idea involves the discretization of one of the variables up to a chosen accuracy level (Teles, J.P., Castro, P.M., Matos, H.A. (2013). Multiparametric disaggregation technique for global optimization of polynomial programming problems. J. Glob. Optim. 55, 227–251), by means of a radix-based numeric representation system, coupled with a residual variable to effectively make its domain continuous. Binary variables are added to the formulation to choose the appropriate digit for each position together with new sets of continuous variables and constraints leading to the transformation of the original mixed-integer non-linear problem into a larger one of the mixed-integer linear programming type. The new underestimation approach can be made as tight as desired and is shown capable of providing considerably better lower bounds than a widely used global optimization solver for a specific class of design problems involving bilinear terms.     

An approximate approach of global optimization for polynomial programming problems

Many methods for solving polynomial programming problems can only find locally optimal solutions. This paper proposes a method for finding the approximately globally optimal solutions of polynomial programs. Representing a bounded continuous variable x_i as the addition of a discrete variable d_i and a small variable ϵ_i, a polynomial term x_ix_i can be expanded as the sum of d_ix_j, d_jϵ_i; and ϵ_iϵ_j. A procedure is then developed to fully linearize d_ix_j and d_je_i, and to approximately linearize ϵ_iϵ_j with an error below a pre-specified tolerance. This linearization procedure can also be extended to higher order polynomial programs. Several polynomial programming examples in the literature are tested to demonstrate that the proposed method can systematically solve these examples to find the global optimum within a pre-specified error.     

Multi-parametric disaggregation technique for global optimization of polynomial programming problems

This paper discusses a power-based transformation technique that is especially useful when solving polynomial optimization problems, frequently occurring in science and engineering. The polynomial nonlinear problem is primarily transformed into a suitable reformulated problem containing new sets of discrete and continuous variables. By applying a term-wise disaggregation scheme combined with multi-parametric elements, an upper/lower bounding mixed-integer linear program can be derived for minimization/maximization problems. It can then be solved to global optimality through standard methods, with the original problem being approximated to a certain precision level, which can be as tight as desired. Furthermore, this technique can also be applied to signomial problems with rational exponents, after a few effortless algebraic transformations. Numerical examples taken from the literature are used to illustrate the effectiveness of the proposed approach.     

线性约束三次规划问题的全局最优性必要条件和最优化算法

讨论了带线性不等式约束三次规划问题的最优性条件和最优化算法. 首先, 讨论了带有线性不等式约束三次规划问题的 全局最优性必要条件. 然后, 利用全局最优性必要条件, 设计了解线性约束三次规划问题的一个新的局部最优化算法(强局部最优化算法). 再利用辅助函数和所给出的新的局部最优化算法, 设计了带有线性不等式约束三 规划问题的全局最优化算法. 最后, 数值算例说明给出的最优化算法是可行的、有效的.     

Weak sharp minima for piecewise linear multiobjective optimization in normed spaces

In a general normed space, we consider a piecewise linear multiobjective optimization problem. We prove that a cone-convex piecewise linear multiobjective optimization problem always has a global weak sharp minimum property. By a counter example, we show that the weak sharp minimum property does not necessarily hold if the cone-convexity assumption is dropped. Moreover, under the assumption that the ordering cone is polyhedral, we prove that a (not necessarily cone-convex) piecewise linear multiobjective optimization problem always has a bounded weak sharp minimum property.     

Optimality conditions for weak efficiency to vector optimization problems with composed convex functions

We consider a convex optimization problem with a vector valued function as objective function and convex cone inequality constraints. We suppose that each entry of the objective function is the composition of some convex functions. Our aim is to provide necessary and sufficient conditions for the weakly efficient solutions of this vector problem. Moreover, a multiobjective dual treatment is given and weak and strong duality assertions are proved.      

Optimality conditions in terms of contingent epiderivatives for strict local Pareto minima in vector optimization problems with constraints

Positivity - In this paper, primal and dual necessary/sufficient optimality conditions are investigated for vector optimization problem with set, cone and equality constraints. We establish some...     

A vector linear programming approach for certain global optimization problems

A large part of the European natural gas imports originates from unstable regions exposed to the risk of supply failure due to economical and political reasons. This has increased the concerns on the security of supply in the European natural gas market. In this paper, we analyze the security of external supply of the Italian gas market that mainly relies on natural gas imports to cover its internal demand. To this aim, we develop an optimization problem that describes the equilibrium state of a gas supply chain where producers, mid-streamers, and final consumers exchange natural gas and liquefied natural gas. Both long-term contracts (LTCs) and spot pricing systems are considered. Mid-streamers are assumed to be exposed to the external supply risk, which is estimated with indicators that we develop starting from those already existing in the literature. In addition, we investigate different degrees of mid-streamers’ flexibility by comparing a situation where mid-streamers fully satisfy the LTC volume clause (“No FLEX” assumption) to a case where the fulfillment of this volume clause is not compulsory (“FLEX” assumption). Our analysis shows that, in the “No FLEX” case, mid-streamers do not significantly change their supplying choices even when the external supply risk is considered. Under this assumption, they face significant profit losses that, instead, disappear in the “FLEX” case when mid-streamers are more flexible and can modify their supply mix. However, the “FLEX” strategy limits the gas availability in the supply chain leading to a curtailment of the social welfare.     

A global optimization algorithm for polynomial programming problems using a Reformulation-Linearization Technique

This paper is concerned with the development of an algorithm to solve continuous polynomial programming problems for which the objective function and the constraints are specified polynomials. A linear programming relaxation is derived for the problem based on a Reformulation Linearization Technique (RLT), which generates nonlinear (polynomial) implied constraints to be included in the original problem, and subsequently linearizes the resulting problem by defining new variables, one for each distinct polynomial term. This construct is then used to obtain lower bounds in the context of a proposed branch and bound scheme, which is proven to converge to a global optimal solution. A numerical example is presented to illustrate the proposed algorithm.