Global Optimality Conditions for Some Classes of Optimization Problems

We establish new necessary and sufficient optimality conditions for global optimization problems. In particular, we establish tractable optimality conditions for the problems of minimizing a weakly convex or concave function subject to standard constraints, such as box constraints, binary constraints, and simplex constraints. We also derive some new necessary and sufficient optimality conditions for quadratic optimization. Our main theoretical tool for establishing these optimality conditions is abstract convexity.     

Global optimality principles for polynomial optimization over box or bivalent constraints by separable polynomial approximations

In this paper we present necessary conditions for global optimality for polynomial problems with box or bivalent constraints using separable polynomial relaxations. We achieve this by first deriving a numerically checkable characterization of global optimality for separable polynomial problems with box as well as bivalent constraints. Our necessary optimality conditions can be numerically checked by solving semi-definite programming problems. Then, by employing separable polynomial under-estimators, we establish sufficient conditions for global optimality for classes of polynomial optimization problems with box or bivalent constraints. We construct underestimators using the sum of squares convex (SOS-convex) polynomials of real algebraic geometry. An important feature of SOS-convexity that is generally not shared by the standard convexity is that whether a polynomial is SOS-convex or not can be checked by solving a semidefinite programming problem. We illustrate the versatility of our optimality conditions by simple numerical examples.     

Non-convex quadratic minimization problems with quadratic constraints: global optimality conditions

In this paper, we first examine how global optimality of non-convex constrained optimization problems is related to Lagrange multiplier conditions. We then establish Lagrange multiplier conditions for global optimality of general quadratic minimization problems with quadratic constraints. We also obtain necessary global optimality conditions, which are different from the Lagrange multiplier conditions for special classes of quadratic optimization problems. These classes include weighted least squares with ellipsoidal constraints, and quadratic minimization with binary constraints. We discuss examples which demonstrate that our optimality conditions can effectively be used for identifying global minimizers of certain multi-extremal non-convex quadratic optimization problems. The work of Z. Y. Wu was carried out while the author was at the Department of Applied Mathematics, University of New South Wales, Sydney, Australia.     

Necessary global optimality conditions for nonlinear programming problems with polynomial constraints

In this paper, we develop necessary conditions for global optimality that apply to non-linear programming problems with polynomial constraints which cover a broad range of optimization problems that arise in applications of continuous as well as discrete optimization. In particular, we show that our optimality conditions readily apply to problems where the objective function is the difference of polynomial and convex functions over polynomial constraints, and to classes of fractional programming problems. Our necessary conditions become also sufficient for global optimality for polynomial programming problems. Our approach makes use of polynomial over-estimators and, a polynomial version of a theorem of the alternative which is a variant of the Positivstellensatz in semi-algebraic geometry. We discuss numerical examples to illustrate the significance of our optimality conditions.     

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.     

Necessary Optimality Conditions for Optimal Control Problems with Nonsmooth Mixed State and Control Constraints

In this paper we study an optimal control problem with nonsmooth mixed state and control constraints. In most of the existing results, the necessary optimality condition for optimal control problems with mixed state and control constraints are derived under the Mangasarian-Fromovitz condition and under the assumption that the state and control constraint functions are smooth. In this paper we derive necessary optimality conditions for problems with nonsmooth mixed state and control constraints under constraint qualifications based on pseudo-Lipschitz continuity and calmness of certain set-valued maps. The necessary conditions are stratified, in the sense that they are asserted on precisely the domain upon which the hypotheses (and the optimality) are assumed to hold. Moreover necessary optimality conditions with an Euler inclusion taking an explicit multiplier form are derived for certain cases.     

Approximate optimality conditions for minimax programming problems

In this article we study non-smooth Lipschitz programming problems with set inclusion and abstract constraints. Our aim is to develop approximate optimality conditions for minimax programming problems in absence of any constraint qualification. The optimality conditions are worked out not exactly at the optimal solution but at some points in a neighbourhood of the optimal solution. For this reason, we call the conditions as approximate optimality conditions. Later we extend the results in terms of the limiting subdifferentials in presence of an appropriate constraint qualification thereby leading to the optimality conditions at the exact optimal point.     

Image Space Analysis for Vector Variational Inequalities with Matrix Inequality Constraints and Applications

In this paper, vector variational inequalities (VVI) with matrix inequality constraints are investigated by using the image space analysis. Linear separation for VVI with matrix inequality constraints is characterized by using the saddle-point conditions of the Lagrangian function. Lagrangian-type necessary and sufficient optimality conditions for VVI with matrix inequality constraints are derived by utilizing the separation theorem. Gap functions for VVI with matrix inequality constraints and weak sharp minimum property for the solutions set of VVI with matrix inequality constraints are also considered. The results obtained above are applied to investigate the Lagrangian-type necessary and sufficient optimality conditions for vector linear semidefinite programming problems as well as VVI with convex quadratic inequality constraints.     

Generalized convexity in non-regular programming problems with inequality-type constraints

Convexity plays a very important role in optimization for establishing optimality conditions. Different works have shown that the convexity property can be replaced by a weaker notion, the invexity. In particular, for problems with inequality-type constraints, Martin defined a weaker notion of invexity, the Karush-Kuhn-Tucker-invexity (hereafter KKT-invexity), that is both necessary and sufficient to obtain Karush-Kuhn-Tucker-type optimality conditions. It is well known that for this result to hold the problem has to verify a constraint qualification, i.e., it must be regular or non-degenerate. In non-regular problems, the classical optimality conditions are totally inapplicable. Meaningful results were obtained for problems with inequality-type constraints by Izmailov. They are based on the 2-regularity condition of the constraints at a feasible point. In this work, we generalize Martin's result to non-regular problems by defining an analogous concept, the 2-KKT-invexity, and using the characterization of the tangent cone in the 2-regular case and the necessary optimality condition given by Izmailov.     

Sufficient global optimality conditions for weakly convex minimization problems

In this paper, we present sufficient global optimality conditions for weakly convex minimization problems using abstract convex analysis theory. By introducing (L,X)-subdifferentials of weakly convex functions using a class of quadratic functions, we first obtain some sufficient conditions for global optimization problems with weakly convex objective functions and weakly convex inequality and equality constraints. Some sufficient optimality conditions for problems with additional box constraints and bivalent constraints are then derived.      

Generalized convexity and efficiency for non-regular multiobjective programming problems with inequality-type constraints

For multiobjective problems with inequality-type constraints the necessary conditions for efficient solutions are presented. These conditions are applied when the constraints do not necessarily satisfy any regularity assumptions, and they are based on the concept of 2-regularity introduced by Izmailov. In general, the necessary optimality conditions are not sufficient and the efficient solution set is not the same as the Karush-Kuhn-Tucker points set. So it is necessary to introduce generalized convexity notions. In the multiobjective non-regular case we give the notion of 2-KKT-pseudoinvex-II problems. This new concept of generalized convexity is both necessary and sufficient to guarantee the characterization of all efficient solutions based on the optimality conditions.     

Optimality conditions with state constraints for semilinear systems determined by operator valued measures

In this paper we develop the necessary conditions of optimality for a class of distributed parameter systems (partial differential equations) determined by operator valued measures and controlled by vector measures. Based on some recent results on existence of optimal controls from the space of vector measures, we develop necessary conditions of optimality for a class of control problems. The main results are the necessary conditions of optimality for problems without state constraints and those with state constraints. Also, a conceptual algorithm along with a brief discussion of its convergence is presented.     

Investigation of regularity conditions in optimal control problems with geometric mixed constraints

We investigate regularity conditions in optimal control problems with mixed constraints of a general geometric type, in which a closed non-convex constraint set appears. A closely related question to this issue concerns the derivation of necessary optimality conditions under some regularity conditions on the constraints. By imposing strong and weak regularity condition on the constraints, we provide necessary optimality conditions in the form of Pontryagin maximum principle for the control problem with mixed constraints. The optimality conditions obtained here turn out to be more general than earlier results even in the case when the constraint set is convex. The proofs of our main results are based on a series of technical lemmas which are gathered in the Appendix.     

A new local and global optimization method for mixed integer quadratic programming problems

In this paper, a new local optimization method for mixed integer quadratic programming problems with box constraints is presented by using its necessary global optimality conditions. Then a new global optimization method by combining its sufficient global optimality conditions and an auxiliary function is proposed. Some numerical examples are also presented to show that the proposed optimization methods for mixed integer quadratic programming problems with box constraints are very efficient and stable.     

When the Karush–Kuhn–Tucker Theorem Fails: Constraint Qualifications and Higher-Order Optimality Conditions for Degenerate Optimization Problems

In this paper, we present higher-order analysis of necessary and sufficient optimality conditions for problems with inequality constraints. The paper addresses the case when the constraints are not assumed to be regular at a solution of the optimization problems. In the first two theorems derived in the paper, we show how Karush–Kuhn–Tucker necessary conditions reduce to a specific form containing the objective function only. Then we present optimality conditions of the Karush–Kuhn–Tucker type in Banach spaces under new regularity assumptions. After that, we analyze problems for which the Karush–Kuhn–Tucker form of optimality conditions does not hold and propose necessary and sufficient conditions for those problems. To formulate the optimality conditions, we introduce constraint qualifications for new classes of nonregular nonlinear optimization. The approach of p-regularity used in the paper can be applied to various degenerate nonlinear optimization problems due to its flexibility and generality.     

Maximum principle for optimal control problems with intermediate constraints

We consider optimal control problems with constraints at intermediate points of the trajectory. A natural technique (propagation of phase and control variables) is applied to reduce these problems to a standard optimal control problem of Pontryagin type with equality and inequality constraints at the trajectory endpoints. In this way we derive necessary optimality conditions that generalize the Pontryagin classical maximum principle. The same technique is applied to so-called variable structure problems and to some hybrid problems. The new optimality conditions are compared with the results of other authors and five examples illustrating their application are presented.     

Necessary Conditions in Nonsmooth Minimization via Lower and Upper Subgradients

The paper concerns first-order necessary optimality conditions for problems of minimizing nonsmooth functions under various constraints in infinite-dimensional spaces. Based on advanced tools of variational analysis and generalized differential calculus, we derive general results of two independent types called lower subdifferential and upper subdifferential optimality conditions. The former ones involve basic/limiting subgradients of cost functions, while the latter conditions are expressed via Fréchet/regular upper subgradients in fairly general settings. All the upper subdifferential and major lower subdifferential optimality conditions obtained in the paper are new even in finite dimensions. We give applications of general optimality conditions to mathematical programs with equilibrium constraints deriving new results for this important class of intrinsically nonsmooth optimization problems.     

Optimization under Generalized Equation Constraints in Asplund Spaces

In this article, necessary optimality conditions for mathematical programming problems under generalized equation constraints problems are studied in Asplund spaces. We consider a very general version of the problem and derive necessary optimality conditions under various hypothesis on the problem data and sacrificing the differentiability assumption.     

Sensitivity of solutions to systems of optimality conditions under the violation of constraint qualifications

A survey is given of old and new results on the sensitivity of solutions to systems of optimality conditions with respect to parametric perturbations. Results of this kind play a key role in subtle convergence analysis of various constrained optimization algorithms. General systems of optimality conditions for problems with abstract constraints, Karush-Kuhn-Tucker systems for mathematical programs, and Lagrange systems for problems with equality constraints are examined. Special attention is given to the cases where the traditional constraint qualifications are violated.     

Efficient solutions and optimality conditions for vector equilibrium problems

Necessary optimality conditions for efficient solutions of unconstrained and vector equilibrium problems with equality and inequality constraints are derived. Under assumptions on generalized convexity, necessary optimality conditions for efficient solutions become sufficient optimality conditions. Note that it is not required here that the ordering cone in the objective space has a nonempty interior.