Second-order necessary optimality conditions for domain optimization problems of the Neumann type

In this paper, we treat a domain optimization problem in which the boundary-value problem is a Neumann problem. In the case where the domain is in a three-dimensional Euclidean space, the first-order and the second-order necessary conditions which the optimal domain must satisfy are derived under a constraint which is the generalization of the requisite of constant volume.Portions of this paper were presented at the 13th IFIP Conference on System Modelling and Optimization, Tokyo, Japan, 1987.     

Second-order necessary conditions in a domain optimization problem

Second-order necessary conditions of the Kuhn-Tucker type for optimality in a domain optimization problem are studied. The second variation, corresponding to a boundary variation, of the solution to a boundary-value problem is shown to exist and is given as the solution of a boundary-value problem of the same type. The boundary data are shown to be given in terms of the solution and the first variation of the solution. From these results, the second variation of the objective function is calculated to derive second-order necessary conditions of the Kuhn-Tucker type.A part of this work was presented under the title of Second Variation and Its Application in a Domain Optimization Problem at the 4th IFAC Symposium on Control of Distributed-Parameter Systems, Los Angeles, California, 1986 and appeared in the Proceedings of the Symposium, Control of Distributed Parameter Systems, Pergamon Press, 1986. The author wishes to express his thanks to Professor Y. Sakawa of Osaka University for his encouragement. The author thanks the referees for critical reading and helpful comments.     

Second-order necessary conditions of the Kuhn-Tucker type under new constraint qualifications

We are concerned with a nonlinear programming problem with equality and inequality constraints. We shall give second-order necessary conditions of the Kuhn-Tucker type and prove that the conditions hold under new constraint qualifications. The constraint qualifications are weaker than those given by Ben-Tal (Ref. 1).The author would like to thank Professor N. Furukawa and the referees for their many valuable comments and helpful suggestions.     

Second-order necessary conditions for domain optimization problems in elastic structures,part 1: Surface traction given as a field

In this paper, domain optimization problems for both linear and nonlinear elastic structures are studied. The first variation and the second variation of the objective function are calculated in terms of the solution, of the first variation of the solution for the primal elastic system, and of the adjoint variables introduced. The adjoint variables obey a (fictitious) linear elastic system in contrast with the nonlinear adjoint systems introduced by Dems and Mróz, and by Dems and Haftka. From these results, the first-order and the second-order necessary conditions that an optimal domain should satisfy are immediately derived.Portions of this paper were presented at the 5th IFAC Symposium on Control of Distributed Parameter Systems, Perpignan, France, 1989. The authors would like to express their sincere thanks to the referees for their critical readings.     

Second-order necessary conditions for domain optimization problems in elastic structures,part 2: Surface traction dependent on the shape

The second-order sensitivity analysis for a domain optimization problem is studied for a linear elastic structure. In the primary elastic structure considered, the surface traction, a part of the boundary conditions, depends not only on the position but also on the shape of the structure. The first variation and the second variation of the objective functional are calculated in terms of the solution, the first variation of the solution for the primal elastic system, and of the adjoint variable introduced. Moreover, the first-order and the second-order necessary optimality conditions are derived for the structure under a hydrostatic pressure. As an illustrative problem, a mean compliance design is treated.     

On necessary extremum conditions for finite-dimensional problems with inequality constraints

The finite-dimensional optimization problem with equality and inequality constraints is examined. The case where the classical regularity condition is violated is analyzed. Necessary second-order extremum conditions are obtained that are stronger versions of some available results.     

Second-order necessary optimality conditions for semilinear elliptic optimal control problems

This paper deals with the necessary optimality conditions for semilinear elliptic optimal control problems with a pure pointwise state constraint and mixed pointwise constraints. By computing the so-called ‘sigma-term’, we obtain the second-order necessary optimality conditions for the problems, which is sharper than some previously established results in the literature. Besides, we give a condition which relaxes the Slater condition and guarantees that the Lagrangian is normalized.     

Necessary optimality conditions in an abnormal optimization problem with equality constraints

An abnormal minimization problem with equality constraints and a finite-dimensional image is examined. Second-order necessary conditions for this problem are given that strengthen previously known results.     

Second-order optimality conditions in set-valued optimization via asymptotic derivatives

In this article we give new second-order optimality conditions in set-valued optimization. We use the second-order asymptotic tangent cones to define second-order asymptotic derivatives and employ them to give the optimality conditions. We extend the well-known Dubovitskii–Milutin approach to set-valued optimization to express the optimality conditions given as an empty intersection of certain cones in the objective space. We also use some duality arguments to give new multiplier rules. By following the more commonly adopted direct approach, we also give optimality conditions in terms of a disjunction of certain cones in the image space. Several particular cases are discussed.     

Second-order conditions in extremal problems. The abnormal points

In this paper we study a minimization problem with constraints and obtain first- and second-order necessary conditions for a minimum. Those conditions - as opposed to the known ones - are also informative in the abnormal case. We have introduced the class of 2-normal constraints and shown that for them the ``gap" between the sufficient and the necessary conditions is as minimal as possible. It is proved that a 2-normal mapping is generic.

     

On necessary optimality conditions in vector optimization problems

Necessary conditions of the multiplier rule type for vector optimization problems in Banach spaces are proved by using separation theorems and Ljusternik's theorem. The Pontryagin maximum principle for multiobjective control problems with state constraints is derived from these general conditions. The paper extends to vector optimization results established in the scalar case by Ioffe and Tihomirov.     

Optimality conditions for piecewiseC 2 nonlinear programming

Several types of finite-dimensional nonlinear programming models are considered in this article. Second-order optimality conditions are derived for these models, under the assumption that the functions involved are piecewiseC ². In rough terms, a real-valued function defined on an open subsetW orR ⁿ is said to be piecewiseC ^k onW if it is continuous onW and if it can be constructed by piecing together onW a finite number of functions of classC ^k .     

Higher-order necessary conditions for infinite and semi-infinite optimization

In this paper, we present a unified theory of first-order and higher-order necessary optimality conditions for abstract vector optimization problems in normed linear spaces. We prove general multiplier rules, from which nearly all known first-order, second-order, and higher-order necessary conditions can be derived. In the last section, we prove higher-order necessary conditions for semi-infinite programming problems.This work was developed within the Forschungsschwerpunkt Dynamische Systeme, Universität Bremen, Bremen, West Germany.The author wishes to thank Prof. Dr. D. Hinrichsen for his helpful remarks and discussions during the preparation of this work.     

Second-order necessary conditions for differential inclusion problems

We study second-order necessary conditions for optimality in the unbounded differential inclusion control problem and recover the accessory problem in optimal control theory.     

On necessary and sufficient conditions in vector optimization

It is shown that some general multiplier rules are necessary conditions for vector optimization in infinite-dimensional spaces. Under additional convexity assumptions, these conditions are sufficient. As an application, the Pontryagin maximum principle for cooperative differential games is examined.The authors are grateful to Professor W. Stadler and the referees of the previous edition of this paper for their valuable remarks and suggestions, which have been very helpful in the preparation of this paper.     

Optimality conditions in a step control problem

First-and second-order necessary optimality conditions are obtained for the control of step systems.     

Second order necessary conditions in set constrained differentiable vector optimization

We state second order necessary optimality conditions for a vector optimization problem with an arbitrary feasible set and an order in the final space given by a pointed convex cone with nonempty interior. We establish, in finite-dimensional spaces, second order optimality conditions in dual form by means of Lagrange multipliers rules when the feasible set is defined by a function constrained to a set with convex tangent cone. To pass from general conditions to Lagrange multipliers rules, a generalized Motzkin alternative theorem is provided. All the involved functions are assumed to be twice Fréchet differentiable. Mathematics subject classification 2000:90C29, 90C46This research was partially supported by Ministerio de Ciencia y Tecnología (Spain), project BMF2003-02194.     

Existence of minimizers and necessary conditions in set-valued optimization with equilibrium constraints

In this paper we study set-valued optimization problems with equilibrium constraints (SOPECs) described by parametric generalized equations in the form 0 ∈ G(x) + Q(x), where both G and Q are set-valued mappings between infinite-dimensional spaces. Such models particularly arise from certain optimization-related problems governed by set-valued variational inequalities and first-order optimality conditions in nondifferentiable programming. We establish general results on the existence of optimal solutions under appropriate assumptions of the Palais-Smale type and then derive necessary conditions for optimality in the models under consideration by using advanced tools of variational analysis and generalized differentiation. Dedicated to Jiří V. Outrata on the occasion of his 60th birthday. This research was partly supported by the National Science Foundation under grants DMS-0304989 and DMS-0603846 and by the Australian Research Council under grant DP-0451168.     

Second-order and related extremality conditions in nonlinear programming

This paper is concerned with the problem of characterizing a local minimum of a mathematical programming problem with equality and inequality constraints. The main object is to derive second-order conditions, involving the Hessians of the functions, or related results where some other curvature information is used. The necessary conditions are of the Fritz John type and do not require a constraint qualification. Both the necessary conditions and the sufficient conditions are given in equivalent pairs of primal and dual formulations.This research was partly supported by Project No. NR-947-021, ONR Contract No. N00014-75-0569, with the Center for Cybernetic Studies, and by the National Science Foundation, Grant No. NSF-ENG-76-10260.     

Second-order necessary and sufficient conditions in nonsmooth optimization

In this paper we generalize and sharpen R.W. Chaney's results on unconstrained and constrained second-order necessary and sufficient optimality conditions [5–7] for general Lipschitz functions without the semismoothness assumptionCorresponding author.