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.     

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.     

Approximate necessary conditions for locally weak Pareto optimality

Necessary conditions for a given pointx ⁰ to be a locally weak solution to the Pareto minimization problem of a vector-valued functionF=(f ₁,...,f _m ),F:XR ^m,XR ^m, are presented. As noted in Ref. 1, the classical necessary condition-conv {Df ₁(x ⁰)|i=1,...,m}T ^*(X, x ⁰) need not hold when the contingent coneT is used. We have proven, however, that a properly adjusted approximate version of this classical condition always holds. Strangely enough, the approximation form>2 must be weaker than form=2.The authors would like to thank the anonymous referee for the suggestions which led to an improved presentation of the paper.     

Refinements of necessary conditions for optimality in nonlinear programming

In this paper, a new set of necessary conditions for optimality is introduced with reference to the differentiable nonlinear programming problem. It is shown that these necessary conditions are sharper than the usual Fritz John ones. A constraint qualification relevant to the new necessary conditions is defined and extensions to the locally Lipschitz case are presented.     

Second order necessary optimality conditions for minimizing a sup-type function

The second derivative of an envelope cannot be expressed only by second derivatives of the constituent functions. By taking account of this fact, we derive new second order necessary optimality conditions for minimization of a sup-type function. The conditions involve an extra term besides the second derivative of the Lagrange function. Furthermore, we will comment on the relationship between the extra term and a kind of second order directional derivative of the sup-type function.     

Necessary optimality conditions for singular controls

A definition of singular controls with respect to components is given that includes, in particular, the conventional definition. On the basis of this definition, new necessary optimiality conditions for singular controls with respect to components are derived for the processes governed by systems of ordinary differential equations.     

Generalized Bellman-Hamilton-Jacobi optimality conditions for a control problem with a boundary condition

In this paper we study conditions for optimality of a deterministic control problem where the state of the system is required to stop at the boundary. Using the Clarke generalized gradient, we refine the classical verification theorem and show that it is not only sufficient but also necessary for optimality. It is also shown that the solution to the generalized Bellman-Jacobi-Hamilton equation involving the Clarke generalized gradient is unique among the class of regular functions.     

On sufficient optimality conditions for a quasiconvex programming problem

Some remarks are made on a paper by Bector, Chandra, and Bector (see Ref. 1) concerning the Fritz John and Kuhn-Tucker sufficient optmality conditions as well as duality theorems for a nonlinear programming problem with a quasiconvex objective function.This research was supported by the Italian Ministry of University Scientific and Technological Research.     

Additional necessary conditions for optimal control with state-variable inequality constraints

It is shown that, when the set of necessary conditions for an optimal control problem with state-variable inequality constraints given by Bryson, Denham, and Dreyfus is appropriately augmented, it is equivalent to the (different) set of conditions given by Jacobson, Lele, and Speyer. Relationships among the various multipliers are given.This work was done at NASA Ames Research Center, Moffett Field, California, under a National Research Council Associateship.     

Realistic modelled optimal control problems in aerospace engineering - a challenge to the necessary optimality

A control problem for a hypersonic space vehicle is used to illustrate the need for a generalization of the necessary optimality conditions in the accurate numerical solution of more realistic models for optimal control problems in aerospace engineering.     

Second-order optimality conditions for minimizing a max-function

We study Chaney's and Ben-Tal-Zowe's second-order directional derivatives with applications in minimization problem for max-functions of the formh(x): = max {f(x, τ); τ ∈T},where T is a compact metricspace. We improve Kawasaki's result on necessary condition for such functions in the minimization problem.     

Necessary optimality conditions of a D.C. set-valued bilevel optimization problem

In this article, we consider a bilevel vector optimization problem where objective and constraints are set valued maps. Our approach consists of using a support function [1–3,14,15,32] together with the convex separation principle for the study of necessary optimality conditions for D.C. bilevel set-valued optimization problems. We give optimality conditions in terms of the strong subdifferential of a cone-convex set-valued mapping introduced by Baier and Jahn 6 Baier, J and Jahn, J. 1999. On subdifferentials of set-valued maps. J. Optim. Theory Appl., 100: 233–240. [Crossref], [Web of Science ®] , [Google Scholar] and the weak subdifferential of a cone-convex set-valued mapping of Sawaragi and Tanino 28 Sawaragi, Y and Tanino, T. 1980. Conjugate maps and duality in multiobjective optimization. J. Optim. Theory Appl., 31: 473–499.  [Google Scholar]. The bilevel set-valued problem is transformed into a one level set-valued optimization problem using a transformation originated by Ye and Zhu 34 Ye, JJ and Zhu, DL. 1995. Optimality conditions for bilevel programming problems. Optimization, 33: 9–27. [Taylor & Francis Online] , [Google Scholar]. An example illustrating the usefulness of our result is also given.     

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.     

Necessary conditions for optimal controls for systems governed by parabolic partial delay-differential equations in divergence form with first boundary conditions

In this paper, we consider the question of necessary conditions for optimality for systems governed by second-order parabolic partial delay-differential equations with first boundary conditions. All the coefficients of the system are assumed bounded measurable and contain controls and delays in their arguments. The second-order parabolic partial delay-differential equation is in divergence form. In Theorem 4.1, we present results on the existence and uniqueness of weak solutions in the sense of Ladyzhenskaya-Solonnikov-Ural'ceva for this class of systems. An integral maximum principle and its point-wise version for the corresponding controlled system are established in Theorem 5.1 and Corollary 5.1, respectively.The authors wish to thank Dr. E. Noussair for his stimulating discussion and valuable comments in the preparation of this paper. Further, they also wish to acknowledge the referee of the paper for his valuable suggestions and comments. The discussion presented in Section 6 is in response to his suggestions.     

Legendre-type optimality conditions for a variational problem with inequality state constraints

, they differ from the Legendre-Clebsch condition. They give information about the Hesse matrix of the integrand at not only inactive points but also active points. On the other hand, since the inequality state constraints can be regarded as an infinite number of inequality constraints, they sometimes form an envelope. According to a general theory [9], one has to take the envelope into consideration when one consider second-order necessary optimality conditions for an abstract optimization problem with a generalized inequality constraint. However, we show that we do not need to take it into account when we consider Legendre-type conditions. Finally, we show that any inequality state constraint forms envelopes except two trivial cases. We prove it by presenting an envelope in a visible form. Received April 18, 1995 / Revised version received January 5, 1998 Published online August 18, 1998     

Constrained control of a nonlinear two point boundary value problem,I

In this paper we consider an optimal control problem for a nonlinear second order ordinary differential equation with integral constraints. A necessary optimality condition in form of the Pontryagin minimum principle is derived. The proof is based on McShane-variations of the optimal control, a thorough study of their behaviour in dependence of some denning parameters, a generalized Green formula for second order ordinary differential equations with measurable coefficients and certain tools of convex analysis.Dedicated to Lothar von Wolfersdorf on the occasion of his 60th birthday     

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.     

Second-order necessary and sufficient optimality conditions for minimizing a sup-type function

In this paper, we give second-order necessary and sufficient optimality conditions for a minimization problem of a sup-type functionS(x)=sup{f(x,t);t T}, whereT is a compact set in a metric space and f is a function defined on ⁿ ×T. Our conditions are stated in terms of the first and second derivatives of f(x, t) with respect tox, and involve an extra term besides the second derivative of the ordinary Lagrange function. The extra term is essential when {f(x,t)} _t forms an envelope. We study the relationship between our results, Wetterling [14], and Hettich and Jongen [6].