Properties of Hamiltonian systems in the Pontryagin maximum principle for economic growth problems

We consider an optimal control problem with a functional defined by an improper integral. We study the concavity properties of the maximized Hamiltonian and analyze the Hamiltonian systems in the Pontryagin maximum principle. On the basis of this analysis, we propose an algorithm for constructing an optimal trajectory by gluing the dynamics of the Hamiltonian systems. The algorithm is illustrated by calculating an optimal economic growth trajectory for macroeconomic data.     

On the use of Hamiltonian and maximized Hamiltonian in nondifferentiable control theory

In optimal control problems involving nondifferentiable functions of the state variable, the adjoint differential inclusion can be formulated by either use of the Hamiltonian or the maximized Hamiltonian. In this paper, we solve a production-employment model in which the latter approach must be utilized, since the former does not enable one to determine the optimal policy.Dedicated to G. LeitmannThe authors gratefully acknowledge useful remarks by S. Jørgensen, J. Levine, A. Luhmer, and P. Michel.     

Sufficient optimality conditions for nonconvex control problems with state constraints

The sufficient optimality conditions of Zeidan for optimal control problems (Refs. 1 and 2) are generalized such that they are applicable to problems with pure state-variable inequality constraints. We derive conditions which neither assume the concavity of the Hamiltonian nor the quasiconcavity of the constraints. Global as well as local optimality conditions are presented.     

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.     

ρ-Invex Functions and ( F , ρ)Convex Functions: Properties and Equivalences

-invexity, -pseudo invexity and -quasi invexity (and their extentions to nondifferentiable Lipschitz functions) have been used to weaken the assumption of convexity in solving duality problems or to state sufficient optimality conditions in nonlinear programming. An attempt to generalize further these concepts has been done with the introduction of ( F , )convexity for differentiable and nondifferentiable Lipschitz functions. Theorems and results regarding both the duality problems and the sufficience of Kuhn-Tucker conditions have been reproduced for these new classes of functions. The aim of this article is to show that for both differentiable and nondifferentiable Lipschitz functions ( F , )convexity is not a generalization of -invexity, but these families of functions coincide.     

Nondifferentiable Minimax Fractional Programming Problems with (C, α, ρ, d)-Convexity

In this paper, we present necessary optimality conditions for nondifferentiable minimax fractional programming problems. A new concept of generalized convexity, called (C, α, ρ, d)-convexity, is introduced. We establish also sufficient optimality conditions for nondifferentiable minimax fractional programming problems from the viewpoint of the new generalized convexity. When the sufficient conditions are utilized, the corresponding duality theorems are derived for two types of dual programs. This research was partially supported by NSF and Air Force grants     

Sufficient optimality conditions in nondifferentiable optimization ∗

The concept of local cone approximations, introduced in recent years by Ester and Thierfelder, is a powerful tool for establishing optimality conditions. In this paper we show how to use it for obtaining sufficient optimality conditions in nondifferentiable optimization     

Optimality conditions for nondifferentiable minimax fractional programming with complex variables

We show that a minimax fractional programming problem is equivalent to a minimax nonfractional parametric problem for a given parameter in complex space. We establish the necessary and sufficient optimality conditions of nondifferentiable minimax fractional programming problem with complex variables under generalized convexities.     

Duality and a Characterization of Pseudoinvexity for Pareto and Weak Pareto Solutions in Nondifferentiable Multiobjective Programming

In this paper, we unify recent optimality results under directional derivatives by the introduction of new pseudoinvex classes of functions, in relation to the study of Pareto and weak Pareto solutions for nondifferentiable multiobjective programming problems. We prove that in order for feasible solutions satisfying Fritz John conditions to be Pareto or weak Pareto solutions, it is necessary and sufficient that the nondifferentiable multiobjective problem functions belong to these classes of functions, which is illustrated by an example. We also study the dual problem and establish weak, strong, and converse duality results.     

Regularity conditions for constrained extremum problems

Necessary and/or sufficient conditions are stated in order to have regularity for nondifferentiable problems or differentiable problems. These conditions are compared with some known constraint qualifications.     

Optimality and Duality for a Class of Nondifferentiable Multiobjective Fractional Programming Problems

In this paper, we consider a class of nondifferentiable multiobjective fractional programs in which each component of the objective function contains a term involving the support function of a compact convex set. We establish necessary and sufficient optimality conditions and duality results for weakly efficient solutions of nondifferentiable multiobjective fractional programming problems. This work was supported by Grant R01-2003-000-10825-0 from the Basic Research Program of KOSEF.     

Parameter-Free Dual Models for Fractional Programming with Generalized Invexity

By parameter-free approach, we establish sufficient optimality conditions for nondifferentiable fractional variational programming under certain specific structure of generalized invexity. Employing the sufficient optimality conditions, two parameter-free dual models are formulated. The weak duality, strong duality and strict converse duality theorems are proved in the framework of generalized invexity.     

An Algorithm Using Trust Region Strategy for Minimization of a Nondifferentiable Function

In this article, we present a method for minimization of a nondifferentiable function. The method uses trust region strategy combined with a bundle method philosophy. It is proved that the sequence of points generated by the algorithm has an accumulation point that satisfies the first order necessary and sufficient conditions.     

Sufficient Stochastic Maximum Principle for Discounted Control Problem

In this article, the sufficient Pontryagin’s maximum principle for infinite horizon discounted stochastic control problem is established. The sufficiency is ensured by an additional assumption of concavity of the Hamiltonian function. Throughout the paper, it is assumed that the control domain \(U\) is a convex bounded set and the control may enter the diffusion term of the state equation. The general results are applied to the controlled stochastic logistic equation of population dynamics.     

Three kinds of convexity and concavity properties of optimal value function in parametric nonlinear programming

This work is concerned with exploring the new convexity and concavity properties of the optimal value function in parametric programming. Some convex (concave) functions are discussed and sufficient conditions for new convexity and concavity of the optimal value function in parametric programming are given. Many results in this paper can be considered as deepen the convexity and concavity studies of convex (concave) functions and the optimal value functions.     

Maximum Principles of Markov Regime-Switching Forward–Backward Stochastic Differential Equations with Jumps and Partial Information

This paper presents three versions of maximum principle for a stochastic optimal control problem of Markov regime-switching forward–backward stochastic differential equations with jumps. First, a general sufficient maximum principle for optimal control for a system, driven by a Markov regime-switching forward–backward jump–diffusion model, is developed. In the regime-switching case, it might happen that the associated Hamiltonian is not concave and hence the classical maximum principle cannot be applied. Hence, an equivalent type maximum principle is introduced and proved. In view of solving an optimal control problem when the Hamiltonian is not concave, we use a third approach based on Malliavin calculus to derive a general stochastic maximum principle. This approach also enables us to derive an explicit solution of a control problem when the concavity assumption is not satisfied. In addition, the framework we propose allows us to apply our results to solve a recursive utility maximization problem.     

On strong KKT optimality conditions for multiobjective semi-infinite programming problems with Lipschitzian data

In this paper, we consider a nondifferentiable multiobjective semi-infinite optimization problem. We introduce a qualification condition and derive strong Karusk Kuhn Tucker(KKT) necessary conditions. Then a sufficient optimality condition is proved under invexity assumptions.     

Necessary and Sufficient Conditions for Minimax Fractional Programming

We establish the necessary and sufficient optimality conditions for a class of nondifferentiable minimax fractional programming problems solving generalized convex functions. Subsequently, we apply the optimality conditions to formulate one parametric dual problem and we prove weak duality, strong duality, and strict converse duality theorems.     

Some results on regularity in vector optimization

In this paper we suggest an approach to regularity in, vector optimization which extends the one given in [9]; some necessary or sufficient regularity conditions are given for a wide class of nondifferentiable vector optimization problems which embraces the convex ones.     

Optimality Conditions and Duality for?Nondifferentiable Multiobjective Fractional Programming Problems with?(C,α,ρ,d)-convexity

The purpose of this paper is to consider a class of nondifferentiable multiobjective fractional programming problems in which every component of the objective function contains a term involving the support function of a compact convex set. Based on the (C,α,ρ,d)-convexity, sufficient optimality conditions and duality results for weakly efficient solutions of the nondifferentiable multiobjective fractional programming problem are established. The results extend and improve the corresponding results in the literature.