Duality theorems for convex problems with time delay

In this paper, we examine a class of convex problems of Bolza type, involving a time delay in the state. It encompasses a variety of time-delay problems arising in the calculus of variations and optimal control. A duality analysis is carried out which, among other things, leads to a characterization of minimizers in terms of the Euler-Lagrange inclusion. The results obtained improve in significant respects on what is achievable by techniques previously employed, based on elimination of the time delay by introduction of an infinite-dimensional state space or on the method of steps.     

Duality theorems for perturbed convex optimization

We show that if F, X are two locally convex spaces and $\text{h: F →}\text{R}\text{?}\text{, ?: F × X → R}$ are two convex functionals satisfying h(y) = ?(y, x₀) (y?F) for some x₀?X, then, under suitable assumptions, the computation of inf h(F) can be reduced to the computation of inf ?(H) on certain hyperplanes H of F × X. We give some applications.     

Duality theorems for marginal problems

Summary Given topological spaces X ₁, ..., X _n with product space X, probability measures _i on X _i together with a real function h on X define a marginal problem as well as a dual problem. Using an extended version of Choquet's theorem on capacities, an analogue of the classical duality theorem of linear programming is established, imposing only weak conditions on the topology of the spaces X _i and the measurability resp. boundedness of the function h. Applications concern, among others, measures with given support, stochastic order and general marginal problems.     

Duality theorems for linear systems and convex systems

We show that, if (F →^uX) is a linear system, $\text{Ω ? X}$ a convex target set and $\text{h: X →}\text{R}\text{?}$ a convex functional, then, under suitable assumptions, the computation of inf $\text{h({y ? F ¦ u(y) ? Ω}}$) can be reduced to the computation of the infimum of h on certain strips or hyperplanes in F, determined by elements of $\text{u}{}^1\text{(X}{}^1\text{)}$, or of the infima on F of Lagrangians, involving elements of $\text{u}{}^1\text{(X}{}^1\text{)}$. Also, we prove similar results for a convex system (F →^uX) and the convex cone Ω of all non-positive elements in X.     

Existence theorems for multidimensional Lagrange problems

Existence theorems are proved for multidimensional Lagrange problems of the calculus of variations and optimal control. The unknowns are functions of several independent variables in a fixed bounded domain, the cost functional is a multiple integral, and the side conditions are partial differential equations, not necessarily linear, with assigned boundary conditions. Also, unilateral constraints may be prescribed both on the space and the control variables. These constraints are expressed by requiring that space and control variables take their values in certain fixed or variable sets wich are assumed to be closed but not necessarily compact.This research was partially supported by the Office of Scientific Research, Office of Aerospace Research, United States Air Force, Grant No. AF-AFOSR-942-65.     

Turnpike theorems for convex problems

We prove turnpike theorems for systems described by differential inclusions with convex graphs.     

Duality theorems in fuzzy optimization problems

Several duality formulation and theorems, and relationship between them in fuzzy programming problems have been studied in literature by various authors under different conditions. In this paper, by considering a partial order relation on the set of fuzzy numbers, and convexity with differentiability of fuzzy mappings, we discuss duality theorems and relationships between them in fuzzy optimization problems with fuzzy coefficients.     

Duality theorems for a class of nonconvex programming problems

The object of this paper is to prove duality theorems for quasiconvex programming problems. The principal tool used is the transformation introduced by Manas for reducing a nonconvex programming problem to a convex programming problem. Duality in the case of linear, quadratic, and linear-fractional programming is a particular case of this general case.The authors are grateful to the referees for their kind suggestions.     

Existence theorems for general control problems of Bolza and Lagrange

The existence of solutions is established for a very general class of problems in the calculus of variations and optimal control involving ordinary differential equations or contingent equations. The theorems, while relatively simple to state, cover, besides the more classical cases, problems with considerably weaker assumptions of continuity or boundedness. For example, the cost functional may only be lower semicontinuous in the control and may approach + ∞ as one nears certain boundary points of the control region; both endpoints in the problem may be “free”. Earlier results of Cesari, Olech and the author are thereby extended.The development is based on the theory of convex integral functionals and their conjugates. The first step is to show that, for purposes of existence theory, the problem can be reduced to a simpler model where control variables are not present as such. This model, resembling a classical problem of Bolza in the calculus of variations, but where the functions are extended-real-valued, is then investigated using, above all, the conjugacy correspondence between generalized Lagrangians and Hamiltonians.     

Lower closure theorems for Lagrange problems of optimization with distributed and boundary controls

Lower closure theorems for Lagrange problems of control have the same role as lower semicontinuity theorems have for free problems of the calculus of variations. In the present paper, we prove lower closure theorems in a form needed for application to optimization problems with distributed and boundary controls. Examples are given with state variables in Sobolev spaces.This work was done in the frame of AFOSR Grant No. 69-1662.     

On strong and total Lagrange duality for convex optimization problems

We give some necessary and sufficient conditions which completely characterize the strong and total Lagrange duality, respectively, for convex optimization problems in separated locally convex spaces. We also prove similar statements for the problems obtained by perturbing the objective functions of the primal problems by arbitrary linear functionals. In the particular case when we deal with convex optimization problems having infinitely many convex inequalities as constraints the conditions we work with turn into the so-called Farkas-Minkowski and locally Farkas-Minkowski conditions for systems of convex inequalities, recently used in the literature. Moreover, we show that our new results extend some existing ones in the literature.     

The infinite dimensional Lagrange multiplier rule for convex optimization problems

In this paper an infinite dimensional generalized Lagrange multipliers rule for convex optimization problems is presented and necessary and sufficient optimality conditions are given in order to guarantee the strong duality. Furthermore, an application is presented, in particular the existence of Lagrange multipliers associated to the bi-obstacle problem is obtained.     

Duality theorems in some overdetermined boundary value problems

We consider several elliptic boundary value problems for which there is an overspecification of data on the boundary of the domain. After reformulating the problems in an equivalent integral form, we use the alternate integral formulation to deduce that if a solution exists, then the domain must be an N-ball. Various Green's functions and classical boundary value problems of second, fourth and higher order are included among the problems considered here.     

Duality for almost convex optimization problems via the perturbation approach

We deal with duality for almost convex finite dimensional optimization problems by means of the classical perturbation approach. To this aim some standard results from the convex analysis are extended to the case of almost convex sets and functions. The duality for some classes of primal-dual problems is derived as a special case of the general approach. The sufficient regularity conditions we need for guaranteeing strong duality are proved to be similar to the ones in the convex case. The research of the first and third authors was partially supported by DFG (German Research Foundation), project WA 922/1. The research of the second author was supported by the grant PN II, ID 523/2007.     

Duality theorems for a new class of multitime multiobjective variational problems

In this work, we consider a new class of multitime multiobjective variational problems of minimizing a vector of functionals of curvilinear integral type. Based on the normal efficiency conditions for multitime multiobjective variational problems, we study duals of Mond-Weir type, generalized Mond-Weir-Zalmai type and under some assumptions of (??, b)-quasiinvexity, duality theorems are stated. We give weak duality theorems, proving that the value of the objective function of the primal cannot exceed the value of the dual. Moreover, we study the connection between values of the objective functions of the primal and dual programs, in direct and converse duality theorems. While the results in §1 and §2 are introductory in nature, to the best of our knowledge, the results in §3 are new and they have not been reported in literature.     

Stable strong Fenchel and Lagrange duality for evenly convex optimization problems

By means of a conjugation scheme based on generalized convex conjugation theory instead of Fenchel conjugation, we build an alternative dual problem, using the perturbational approach, for a general optimization one defined on a separated locally convex topological space. Conditions guaranteeing strong duality for primal problems which are perturbed by continuous linear functionals and their respective dual problems, which is named stable strong duality, are established. In these conditions, the fact that the perturbation function is evenly convex will play a fundamental role. Stable strong duality will also be studied in particular for Fenchel and Lagrange primal–dual problems, obtaining a characterization for Fenchel case.     

Duality and optimality for convex extremal problems described by discrete inclusions

This paper is devoted to the Hamiltonian approach for extremal problems concerning convex (multi-valued) mapping. The approach exploits the concept of a Hamiltonian function permitting simplified proofs and useful mathematical insights. Moreover it provides in a duality framework a common point ox view upon the methods used. by Rockafellar, (the theory of convex processes), Pshenichnyi (the conjugate transformation method) and CASS (the symmetric duality scheme) to construct optimality conditions. The theory is used to develop a complete characterization of optimal solutions for multi-period convex programming problems.     

Saddle points of hamiltonian systems in convex problems of Lagrange

In Lagrange problems of the calculus of variations where the LagrangianL(x ), not necessarily differentiable, is convex jointly inx and , optimal arcs can be characterized in terms of a generalized Hamiltonian differential equation, where the HamiltonianH(x, p) is concave inx and convex inp. In this paper, the Hamiltonian system is studied in a neighborhood of a minimax saddle point ofH. It is shown under a strict concavity-convexity assumption onH that the point acts much like a saddle point in the sense of differential equations. At the same time, results are obtained for problems in which the Lagrange integral is minimized over an infinite interval. These results are motivated by questions in theoretical economics.This research was supported in part by Grant No. AFOSR-71-1994.     

集合函数多目标规划的拉格朗日型对偶定理

本文在［１］的基础上．给出了集合函数多目标规划的拉格朗日型弱对偶定理，严格对偶定理和逆对偶定理．