First order optimality conditions for mathematical programs with semidefinite cone complementarity constraints

In this paper we consider a mathematical program with semidefinite cone complementarity constraints (SDCMPCC). Such a problem is a matrix analogue of the mathematical program with (vector) complementarity constraints (MPCC) and includes MPCC as a special case. We first derive explicit formulas for the proximal and limiting normal cone of the graph of the normal cone to the positive semidefinite cone. Using these formulas and classical nonsmooth first order necessary optimality conditions we derive explicit expressions for the strong-, Mordukhovich- and Clarke- (S-, M- and C-)stationary conditions. Moreover we give constraint qualifications under which a local solution of SDCMPCC is a S-, M- and C-stationary point. Moreover we show that applying these results to MPCC produces new and weaker necessary optimality conditions.     

Sufficient conditions for preference optimality

Preference optimality is an optimality concept in multicriteria problems, that is, in problems where several criteria are to beoptimized simultaneously. Formally, one wishes to optimizeN criteriag _i(·) or, equivalently, a criterion vectorg(·) ^N , subject to either functional constraints in programming or to side conditions which are differential equations in optimal control. Subject to these constraints, one obtains forg(·) a set of attainable values in ^N . This set is preordered by the introduction of a complete preordering ; a controlu*(·) or a decisionx*, then, is preference-optimal if it results ing(u*(·))g(u(·)) for all admissible controlsu(·) or ifg(x*)g(x) for all feasible decisionsx. The present paper concerns sufficient conditions for preference-optimal control and for preference-optimal decisions.     

Necessary optimality conditions and exact penalization for non-Lipschitz nonlinear programs

Mathematical Programming - When the objective function is not locally Lipschitz, constraint qualifications are no longer sufficient for Karush–Kuhn–Tucker (KKT) conditions to hold at a...     

Sufficient optimality conditions for some structural design problems

Summary The derivation of sufficient conditions for optimal structural design is considered in this paper. It is shown how such conditions can be obtained for certain classes of problems. Examples involving design constraints on deflection and stability are presented to demonstrate the procedure.

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Zusammenfassung Die Arbeit befaßt sich mit der Herleitung hinreichender Bedingungen für optimale Dimensionierung. Es wird gezeigt, wie für gewisse Problemklassen solche Bedingungen gewonnen werden können. Das Verfahren wird an Beispielen erläutert, welche Bedingungen bezüglich Deformation und Stabilität unterliegen.

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This research was supported by the U.S. Army Research Office, Durham.     

Sufficient second-order optimality conditions for convex control constraints

In this article sufficient optimality conditions are established for optimal control problems with pointwise convex control constraints. Here, the control is a function with values in Rn. The constraint is of the form u(x)∈U(x), where U is a set-valued mapping that is assumed to be measurable with convex and closed images. The second-order condition requires coercivity of the Lagrange function on a suitable subspace, which excludes strongly active constraints, together with first-order necessary conditions. It ensures local optimality of a reference function in an L^∞-neighborhood. The analysis is done for a model problem namely the optimal distributed control of the instationary Navier-Stokes equations.     

多目标半定规划的最优性条件及对偶理论

在不变凸的假设下来讨论多目标半定规划的最优性条件、对偶理论以及非凸半定规划的最优性条件．首先给出了非凸半定规划的一个KKT条件成立的充分必要条件, 并利用此定理证明了其最优性必要条件．其次讨论了多目标半定规划的最优性必要条件、充分条件, 并对其建立Wolfe对偶模型, 证明了弱对偶定理和强对偶定理．     

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.      

Sufficient conditions for optimality in problems with state constraints

In this paper the sufficient conditions for optimality are obtained for problems with state constraints. These constraints may be active. It means that the adjoint function may have points of discontinuity or jumps. Similar results in the case of absolutely continuous adjoint function were given by the author in [1] and [2].     

Sufficient optimality criteria in nonlinear programming for generalized equality-inequality constraints

Sufficient optimality criteria of the Kuhn-Tucker and Fritz John type in nonlinear programming are established in the presence of equality-inequality constraints. The constraint functions are assumed to be quasiconvex, and the objective function is taken to be pseudoconvex (or convex).     

Sufficient conditions for the almost layer finiteness of groups

We prove a theorem that describes almost layer-finite groups in the class of conjugatively biprimitive-finite groups. Computer Center of the Siberian Division of the Russian Academy of Sciences, Krasnoyarsk. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 4, pp. 472–485, April, 1999.     

Global optimality conditions for fixed charge quadratic programs

In this note, we establish sufficient and necessary global optimality conditions for fixed charge quadratic programming problem. The main theoretical tool for establishing these global optimality conditions is abstract convexity. The newly obtained sufficient condition extends the existing sufficient conditions. A numerical example is also provided to illustrate our optimality conditions.     

Dual conditions characterizing optimality for convex multi-objective programs

Received June 13, 1995 / Revised version received February 6, 1998 Published online August 18, 1998     

Sufficient optimality conditions for nonconvex control problems with state constraints

The sufficient optimality conditions of Zeidan for optimal control problems (Refs. 1 and 2) are generalized such that they are applicable to problems with pure state-variable inequality constraints. We derive conditions which neither assume the concavity of the Hamiltonian nor the quasiconcavity of the constraints. Global as well as local optimality conditions are presented.     

Sufficient optimality conditions and duality theory for interval optimization problem

This paper addresses the duality theory of a nonlinear optimization model whose objective function and constraints are interval valued functions. Sufficient optimality conditions are obtained for the existence of an efficient solution. Three type dual problems are introduced. Relations between the primal and different dual problems are derived. These theoretical developments are illustrated through numerical example.     

Sufficient optimality conditions and duality for a quasiconvex programming problem

Under differentiability assumptions, Fritz John Sufficient optimality conditions are proved for a nonlinear programming problem in which the objective function is assumed to be quasiconvex and the constraint functions are assumed to quasiconcave/strictly pseudoconcave. Duality theorems are proved for Mond-Weir type duality under the above generalized convexity assumptions.The first author is thankful to the Natural Science and Engineering Research Council of Canada for financial support through Grant No. A-5319. The authors are thankful to Professor B. Mond for suggestions that improved the original draft of the paper.     

Optimality conditions and global convergence for nonlinear semidefinite programming

Mathematical Programming - Sequential optimality conditions have played a major role in unifying and extending global convergence results for several classes of algorithms for general nonlinear...     

Sufficient optimality conditions for extremal controls based on functional increment formulas

We consider the optimal control problem without terminal constraints. With the help of nonstandard functional increment formulas we introduce definitions of strongly extremal controls. Such controls are optimal in linear and quadratic problems. In the general case, the optimality property is provided with an additional concavity condition of Pontryagin’s function with respect to phase variables.     

Sufficient conditions for optimality of threat strategies in a differential game

The problem of defining threat strategies in nonzero-sum games is considered, and a definition of optimal threat strategies is proposed in the static case. This definition is then extended to differential games, and sufficient conditions for optimality of threat strategies are derived. These are then applied to a simple example. The definition proposed here is then compared with the definition of threat strategies given by Nash.     

Sufficient conditions for optimality for differential inclusions of parabolic type and duality

Sufficient conditions for optimality are derived for partial differential inclusions of parabolic type on the basis of the apparatus of locally conjugate mapping, and duality theorems are proved. The duality theorems proved allow one to conclude that a sufficient condition for an extremum is an extremal relation for the direct and dual problems.