Necessary and sufficient conditions for discrete and differential inclusions of elliptic type

This paper deals for the first time with the Dirichlet problem for discrete (PD), discrete approximation problem on a uniform grid and differential (PC) inclusions of elliptic type. In the form of Euler-Lagrange inclusion necessary and sufficient conditions for optimality are derived for the problems under consideration on the basis of new concepts of locally adjoint mappings. The results obtained are generalized to the multidimensional case with a second order elliptic operator.     

Necessary and sufficient conditions for linear suboptimality in constrained optimization

This article is devoted to the study of some extremality and optimality notions that are different from conventional concepts of optimal solutions to optimization-related problems. These notions reflect certain amounts of linear subextremality for set systems and linear suboptimality for feasible solutions to multiobjective and scalar optimization problems. In contrast to standard notions of optimality, it is possible to derive necessary and sufficient conditions for linear subextremality and suboptimality in general nonconvex settings, which is done in this article via robust generalized differential constructions of variational analysis in finite-dimensional and infinite-dimensional spaces.      

Necessary conditions for super minimizers in constrained multiobjective optimization

This paper concerns the study of the so-called super minimizers related to the concept of super efficiency in constrained problems of multiobjective optimization, where cost mappings are generally set-valued. We derive necessary conditions for super minimizers on the base of advanced tools of variational analysis and generalized differentiation that are new in both finite-dimensional and infinite-dimensional settings for problems with single-valued and set-valued objectives.      

Necessary conditions for stochastic dominance: A general approach

In this paper a general method for developing necessary conditions for all degrees of stochastic dominance is derived. The method, a minimization of the expected value of certain functions of the random variable, is used to rederive known necessary conditions for dominance and is then used to derive new necessary conditions. Some of the old and new conditions are then compared empirically using a data set of security returns.     

Necessary conditions for distributed-parameter systems with controls of fewer variables than state variables

Necessary conditions are proved for multi-dimensional control systems, in which the state is a function of several independent variables in a fixed domain, while the controlu is a function of only some of the independent variables (that is,u is a lower-dimensional control). The cost functional can be taken in the usual Lagrange form of a multiple integral or in Mayer form. The state equations are a system of partial differential equations in explicit form with assigned boundary conditions, and constraints may be imposed on the values of the control variables.The author wishes to thank Professor L. Cesari for his many helpful comments and assistance in the preparation of this paper and Professor L. DeBoer for his interest and encouragement. This work was sponsored by the United States Air Force under Grants Nos. AF-AFOSR-69-1767-A and AFOSR-69-1662.     

Necessary Optimality Conditions for a Dead Oil Isotherm Optimal Control Problem

We study a system of nonlinear partial differential equations resulting from the traditional modelling of oil engineering within the framework of the mechanics of a continuous medium. Recent results on the problem provide existence, uniqueness and regularity of the optimal solution. Here we obtain the first necessary optimality conditions. Work supported by the Portuguese Foundation for Science and Technology (FCT) through the Centre for Research in Optimization and Control (CEOC) of the University of Aveiro, cofinanced by the European Community fund FEDER/POCTI. The first author was also supported by the postdoctoral fellowship SFRH/BPD/20934/2004.     

Necessary conditions for optimal control problems with state constraints

Necessary conditions of optimality are derived for optimal control problems with pathwise state constraints, in which the dynamic constraint is modelled as a differential inclusion. The novel feature of the conditions is the unrestrictive nature of the hypotheses under which these conditions are shown to be valid. An Euler Lagrange type condition is obtained for problems where the multifunction associated with the dynamic constraint has values possibly unbounded, nonconvex sets and satisfies a mild `one-sided' Lipschitz continuity hypothesis. We recover as a special case the sharpest available necessary conditions for state constraint free problems proved in a recent paper by Ioffe. For problems where the multifunction is convex valued it is shown that the necessary conditions are still valid when the one-sided Lipschitz hypothesis is replaced by a milder, local hypothesis. A recent `dualization' theorem permits us to infer a strengthened form of the Hamiltonian inclusion from the Euler Lagrange condition. The necessary conditions for state constrained problems with convex valued multifunctions are derived under hypotheses on the dynamics which are significantly weaker than those invoked by Loewen and Rockafellar to achieve related necessary conditions for state constrained problems, and improve on available results in certain respects even when specialized to the state constraint free case.

Proofs make use of recent `decoupling' ideas of the authors, which reduce the optimization problem to one to which Pontryagin's maximum principle is applicable, and a refined penalization technique to deal with the dynamic constraint.

     

Necessary and sufficient optimality conditions for mathematical programs with equilibrium constraints

In this paper we consider a mathematical program with equilibrium constraints (MPEC) formulated as a mathematical program with complementarity constraints. Various stationary conditions for MPECs exist in literature due to different reformulations. We give a simple proof to the M-stationary condition and show that it is sufficient for global or local optimality under some MPEC generalized convexity assumptions. Moreover, we propose new constraint qualifications for M-stationary conditions to hold. These new constraint qualifications include piecewise MFCQ, piecewise Slater condition, MPEC weak reverse convex constraint qualification, MPEC Arrow-Hurwicz-Uzawa constraint qualification, MPEC Zangwill constraint qualification, MPEC Kuhn-Tucker constraint qualification, and MPEC Abadie constraint qualification.     

Solvability of nonlinear impulsive Volterra integral inclusions and functional differential inclusions

This paper presents sufficient conditions for the existence of solutions to nonlinear impulsive Volterra integral inclusions and initial value problems for second order impulsive functional differential inclusions in Banach spaces. Our results are obtained via a fixed point theorem due to Hong [J. Math. Anal. Appl. 282 (2003) 151-162] for discontinuous multivalued increasing operators.     

Necessary conditions for optimal control problems with bounded state by a penalty method

Necessary conditions are derived for a general relaxed control problem with unilateral state constraint. The results are also valid for ordinary controls that are solutions of the relaxed problem.A penalty is imposed to change the constrained problem into a sequence of unconstrained problems. The assumptions are on the data of the problem and do not requirea priori verification of hypotheses involving the optimal solution.     

Deterministic near-optimal control,part 1: Necessary and sufficient conditions for near-optimality

Near-optimization is as sensible and important as optimization for both theory and applications. This paper concerns dynamic near-optimization, or near-optimal control, for systems governed by deterministic ordinary differential equations. Necessary and sufficient conditions for near-optima control are studied. It is shown that any near-optimal control nearly maximizes the Hamiltonian in some integral sense, and vice versa, if some additional concavity conditions are imposed. Error estimates for both the near-optimality of the controls and the near-maximality of the Hamiltonian are obtained. A number of examples are presented to illustrate these results.This work was supported by the RGC Earmarked Grant CUHK 249/94E. Helpful comments from L. D. Berkovitz are gratefully acknowledged.     

Necessary and sufficient conditions in constrained optimization

Additional conditions are attached to the Kuhn-Tucker conditions giving a set of conditions which are both necessary and sufficient for optimality in constrained optimization, under appropriate constraint qualifications. Necessary and sufficient conditions are also given for optimality of the dual problem. Duality and converse duality are treated accordingly.     

Necessary optimality conditions for bilevel set optimization problems

Bilevel programming problems are hierarchical optimization problems where in the upper level problem a function is minimized subject to the graph of the solution set mapping of the lower level problem. In this paper necessary optimality conditions for such problems are derived using the notion of a convexificator by Luc and Jeyakumar. Convexificators are subsets of many other generalized derivatives. Hence, our optimality conditions are stronger than those using e.g., the generalized derivative due to Clarke or Michel-Penot. Using a certain regularity condition Karush-Kuhn-Tucker conditions are obtained.      

Necessary conditions for the neutral problem of Bolza with continuously varying time delay

This note uses Clarke's decoupling technique to obtain necessary conditions for the generalized problem of Bolza with Lipschitz continuously varying delay in both the state and velocity variables.     

Necessary conditions and non-existence results for autonomous nonconvex variational problems

Classical one-dimensional, autonomous Lagrange problems are considered. In absence of any smoothness, convexity or coercivity condition on the energy density, we prove a DuBois-Reymond type necessary condition, expressed as a differential inclusion involving the subdifferential of convex analysis. As a consequence, a non-existence result is obtained.     

Necessary and sufficient conditions for nonsmooth mathematical programs with equilibrium constraints

In this paper we consider a mathematical program with equilibrium constraints (MPEC) formulated as a mathematical program with complementarity constraints. Then, we derive a necessary optimality result for nonsmooth MPEC on any Asplund space. Also, under generalized convexity assumptions, we establish sufficient optimality conditions for this program in Banach spaces.     

Necessary and sufficient conditions for stabilization of discrete-time planar switched systems

This paper discusses the stabilization problem for single-input discrete-time planar switched systems. We establish necessary and sufficient conditions for the stabilization of planar switched systems. A series of linear inequalities are presented for describing the set of all common quadratic Lyapunov functions. Our results are not only easily testable, but also constructive.     

Viable control for uncertain nonlinear dynamical systems described by differential inclusions

In this paper, we will study the viable control problem for a class of uncertain nonlinear dynamical systems described by a differential inclusion. The goal is to construct a feedback control such that all trajectories of the system are viable in a map. Moreover, for any initial states no viable in the map, under the feedback control, all solutions of the system are steered to the map with an exponential convergence rate and viable in the map after a finite time T. In this case, an estimate of the time T of all trajectories attaining the map is given. In the nanomedicine system, an example inspired from cerebral embolism and cerebral thrombosis problems illustrates the use of our main results.