Parameter-free convex equivalent and dual programs of fractional programming problems

Summary An appropriate generalization ofCharnes-Cooper's [1962] variable transformation is introduced, by which a parameter-free convex program is associated to nonlinear fractional programs. The equivalent program also enables a direct approach toJagannathan's. [1973] duality theory simultaneously extending it. In particular for some special cases further duality theorems are derived.

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Zusammenfassung Durch geeignete Verallgemeinerung der Variablentransformation vonCharnes-Cooper [1962] wird nichtlinearen Quotientenprogrammen ein parameterfreies konvexes Programm zugeordnet. Dieses ermöglicht auch einen direkten Zugang zuJagannathans [1973] Dualitätstheorie, die gleichzeitig erweitert wird. Insbesondere werden für einige Spezialfälle weitere Dualitätssätze abgeleitet.

     

Differential stability of solutions to convex,control constrained optimal control problems

A family of convex, control constrained optimal control problems that depend on a real parameter is considered. It is shown that under some regularity conditions on data the solutions of these problems, as well as the associated Lagrange multipliers are directionally differentiable with respect to parameter. The respective right-derivatives are given as the solution and the associated Lagrange multipliers for some quadratic optimal control problem. If a condition of strict complementarity type hold, then directional derivatives become continuous ones.     

Relaxation in nonconvex optimal control problems with subdifferential operators

We consider the minimization problem of an integral functional in a separable Hilbert space with integrand not convex in the control defined on solutions of the control system described by nonlinear evolutionary equations with mixed nonconvex constraints. The evolutionary operator of the system is the subdifferential of a proper, convex, lower semicontinuous function depending on time. Along with the initial problem, the author considers the relaxed problem with the convexicated control constraint and the integrand convexicated with respect to the control. Under sufficiently general assumptions, it is proved that the relaxed problem has an optimal solution, and for any optimal solution, there exists a minimizing sequence of the initial problem converging to the optimal solution with respect to trajectories and the functional. An example of a controlled parabolic variational inequality with obstacle is considered in detail. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 26, Nonlinear Dynamics, 2005.     

Relaxation techniques for strictly convex network problems

Gauss—Seidel type relaxation techniques are applied in the context of strictly convex pure networks with separable cost functions. The algorithm is an extension of the Bertsekas—Tseng approach for solving the linear network problem and its dual as a pair of monotropic programming problems. The method is extended to cover the class of generalized network problems. Alternative internal tactics for the dual problem are examined. Computational experiments —aimed at the improved efficiency of the algorithm — are presented. This research was supported in part by National Science Foundation Grant No. DCR-8401098-A0l.     

Relaxation techniques for strictly convex network problems

Gauss—Seidel type relaxation techniques are applied in the context of strictly convex pure networks with separable cost functions. The algorithm is an extension of the Bertsekas—Tseng approach for solving the linear network problem and its dual as a pair of monotropic programming problems. The method is extended to cover the class of generalized network problems. Alternative internal tactics for the dual problem are examined. Computational experiments — aimed at the improved efficiency of the algorithm — are presented.This research was supported in part by National Science Foundation Grant No. DCR-8401098-A01.     

Relaxation in nonconvex optimal control problems described by fractional differential equations

We consider the minimization problem of an integral functional with integrand that is not convex in the control on solutions of a control system described by fractional differential equation with mixed nonconvex constraints on the control. A relaxation problem is treated along with the original problem. It is proved that, under general assumptions, the relaxation problem has an optimal solution, and that for each optimal solution there is a minimizing sequence of the original problem that converges to the optimal solution with respect to the trajectory, the control, and the functional in appropriate topologies simultaneously.     

On global maximum of a convex terminal functional in optimal control problems

In this paper necessary and sufficient conditions (related to Pontryagin's principle) for a global maximum of a convex terminal functional for different types of control systems are proved. A few examples are given.     

Characterizations of optimal solution sets of convex infinite programs

In this paper, several Lagrange multiplier characterizations of the solution set of a convex infinite programming problem are given. Characterizations of solution sets of cone-constrained convex programs are derived as well. The procedure is then adopted to a semi-convex problem with convex constraints. For this problem, we present firstly a necessary and sufficient condition for optimality and secondly a characterization of its optimal solution set, based on a Lagrange multiplier associated with a given solution and on directional derivatives of the objective function.      

Differential sensitivity of solutions of convex constrained optimal control problems for discrete systems

A family of optimal control problems for discrete systems that depend on a real parameter is considered. The problems are strongly convex and subject to state and control constraints. Some regularity conditions are imposed on the constraints.The control problems are reformulated as mathematical programming problems. It is shown that both the primal and dual optimal variables for these problems are right-differentiable functions of a parameter. The right-derivatives are characterized as solutions to auxiliary quadratic control problems. Conditions of continuous differentiability are discussed, and some estimates of the rate of convergence of the difference quotients to the respective derivatives are given.     

On differentiability with respect to parameter of solutions to convex optimal control problems subject to state space constraints

A family of convex optimal control problems that depend on a real parameterh is considered. The optimal control problems are subject to state space constraints.It is shown that under some regularity conditions on data the solutions of these problems as well as the associated Lagrange multipliers are directionally-differentiable functions of the parameter.The respective right-derivatives are given as the solution and respective Lagrange multipliers for an auxiliary quadratic optimal control problem subject to linear state space constraints.If a condition of strict complementarity type holds, then directional derivatives become continuous ones.     

Singular perturbations in a class of problems of optimal control with integral convex criterion

The problem of optimal control is investigated with a linear law of motion and convex quality criterion. A small positive parameter appears in front of the derivatives of some of the unknowns in the law of motion. The behaviour of the optimal solution is studied when the small parameter approaches zero with some assumptions that are different from thos encountered in the literature.     

Reduction of delayed optimal control problems to nondelayed problems

Typographical errors are corrected in Ref. 1.     

A posteriori error estimates for mixed finite element solutions of convex optimal control problems

In this paper, we present an a posteriori error analysis for mixed finite element approximation of convex optimal control problems. We derive a posteriori error estimates for the coupled state and control approximations under some assumptions which hold in many applications. Such estimates can be used to construct reliable adaptive mixed finite elements for the control problems.     

Reduction of delayed optimal control problems to nondelayed problems

A method is exhibited which transforms a large class of optimal control problems with fixed delays to nondelayed problems, thus permitting classical results to be used in their analysis.     

Pointwise maximum principle for convex optimal control problems with mixed control-phase variable inequality constraints

The validity of a global pointwise maximum principle is proved for a class of convex optimal control problems with mixed control-phase variable inequality constraints. No compatibility hypotheses are required, and singular multipliers are avoided.     

On duality in problems of optimal control described by convex differential inclusions of Goursat-Darboux type

Sufficient conditions of optimality are derived for convex and non-convex problems with state constraints on the basis of the apparatus of locally conjugate mappings. The duality theorem is formulated and the conditions under which the direct and dual problems are connected by the duality relation are searched for.     

A non-well-posed problem in convex optimal control

This article is devoted to the study of a non-well-set problem of optimal control for a system governed by an elliptic partial differential equation. By generalizing some classical methods in convex optimization, we state and prove the system of necessary and sufficient conditions for such a problem.     

Error analysis for convex separable programs: Bounds on optimal and dual optimal solutions

Computable lower and upper bounds on the optimal and dual optimal solutions of a nonlinear, convex separable program are obtained from its piecewise linear approximation. They provide traditional error and sensitivity measures and are shown to be attainable for some problems. In addition, the bounds on the solution can be used to develop an efficient solution approach for such programs, and the dual bounds enable us to determine a subdivision interval which insures the objective function accuracy of a prespecified level. A generalization of the bounds to certain separable, but nonconvex, programs is given and some numerical examples are included.