Continuous-time Programming Problems Research Articles

Continuous time non-smooth optimization through quasi efficiency

Abstract The importance of quasi efficiency lies in its versatile nature as it permits a definite tolerable error that depend on the decision variables. This has been a motivating factor for us to introduce the notion of quasi e cient solution for the non-smooth multiobjective continuous time programming problem. Necessary optimality conditions are derived for this problem. To derive sufficient optimality conditions, the concept of approximate convexity has been extended to continuous case in this paper. A mixed dual is proposed for which weak and strong duality results are proved.

Optimality conditions for isoperimetric continuous - time optimization problems

In this paper we deal with a nonsmooth case of isoperimetric convex continuoustime programming problem with inequality integral constraint and phase constraint, defined in L?([0, T],Rn). In order to obtain necessary optimality conditions for this problem, we will use a theorem of the alternative from [15] as a main tool.

Saddle-point type optimality criteria, duality and a new approach for solving nonsmooth fractional continuous-time programming problems

<p style='text-indent:20px;'>In this paper, fractional continuous-time programming problems with inequality phase constraints are considered. Optimality conditions and duality results under a certain regularity condition are derived. All functions are assumed to be nondifferentiable. These results improve and generalize a number of existing results in the area of fractional continuous-time programming. We provide a practical example to illustrate our results.</p>

Saddle point optimality criteria and duality for convex continuous-time programming problem

In this paper, convex continuous-time programming problem with inequality type of constraints is considered. We derive new saddle point optimality conditions and classical duality results such as weak and strong duality properties, under additional regularity assumption. A fundamental tool, employed in the derivation of the necessary saddle point optimality criteria and strong duality result for convex continuous-time programming, is a new version of a theorem of the alternative in infinite-dimensional spaces.

New optimality criteria for convex continuous-time problems of vector optimization

In this paper, vector continuous-time programming problems with inequality type of constraints are considered. All the results are obtained under convexity assumptions. The main tool employed in the derivation of the zero order optimality conditions is a new version of theorem of the alternative in functional spaces. To apply the alternative theorem, a certain regularity condition must be satisfied. No differentiability assumption is imposed.

New optimality conditions in vector continuous-time programming

In this work vector continuous-time programming problem with inequality constraints is considered. The necessary and sufficient optimality conditions under generalized concavity assumptions are established. The results were formulated using differentiability.

Optimality in continuous-time multiobjective optimization and vector variational-like inequalities

In this paper, we study the relationships between the Stampacchia and Minty vector variational inequalities and vector continuous-time programming problems under generalized invexity and monotonicity hypotheses. We extend the results given by other authors for the scalar case to vectorial one, and we show the equivalence of efficient and weak efficient solutions for the multiobjective continuous-time programming problem, and solutions of Stampacchia and Minty variational-like type inequality problems. Furthermore, we introduce a new concept of generalized invexity for continuous-time programming problems, namely, the Karush–Kuhn–Tucker (KKT)-pseudoinvexity-II. We prove that this new concept is a necessary and sufficient condition for a vector KKT solution to be an efficient solution for a multiobjective continuous-time programming problem. In this work, duality results for Mond–Weir type dual problems are obtained, using generalized KKT-pseudoinvexity-II. This paper gives an unified point of view for optimality results in mathematical programming or control or variational inequalities problems.

Vector continuous-time programming without differentiability

In this work continuous-time programming problems of vector optimization are considered. Firstly, a nonconvex generalized Gordan’s transposition theorem is obtained. Then, the relationship with the associated weighting scalar problem is studied and saddle point optimality results are established. A scalar dual problem is introduced and duality theorems are given. No differentiability assumption is imposed.

Application of continuous-time programming problems to a class of variational-type inequalities

In this paper, we establish sufficiency criteria under generalized ρ − ( η , θ ) -invexity conditions for general continuous-time programming problems with nonlinear equality/inequality constraints. Using this we establish some existence criteria for solutions of a class of variational-type inequalities.

OPTIMALITY CONDITIONS AND DUALITY MODELS FOR A CLASS OF NONSMOOTH CONTINUOUS-TIME GENERALIZED FRACTIONAL PROGRAMMING PROBLEMS

Both parametric and parameter-free stationary-point-type and saddle-point-type necessary and sufficient optimality conditions are established for a class of nonsmooth continuous-time generalized fractional programming problems with Volterra-type integral inequality and nonnegativity constraints. These optimality criteria are then utilized for constructing ten parametric and parameter-free Wolfe-type and Lagrangian-type dual problems and for proving weak, strong, and strict converse duality theorems. Furthermore, it is briefly pointed out how similar optimality and duality results can be obtained for two important special cases of the main problem containing arbitrary norms and square roots of positive semidefinite quadratic forms. All the results developed here are also applicable to continuous-time programming problems with fractional, discrete max, and conventional objective functions, which are special cases of the main problem studied in this paper.

Proper efficiency principles and duality models for a class of continuous-time multiobjective fractional programming problems with operator constraints

Necessary and sufficient saddle-point- and stationary-point-type proper efficiency conditions are established for a class of continuous-time multiobjective fractional programming problems with convex operator inequality and affine operator equality constraints. Subsequently, the form and contents of these proper efficiency results are utilized for constracting a Lagrangian-type, two parametric, and four semiparametric duality models, and for proving appropriate duality theorems. These proper efficiency and duality results contain, as special cases, similar results for continuous-time programming problems with nonfractional multiple, single fractional, and conventional objective functions, which are patricular cases of the principal problem studied in this paper.

Optimality conditions and duality models for a class of nonsmooth continuous-time fractional programming problems

Both parametric and nonparametric necessary and sufficient optimality conditions are established for a class of nonsmooth continuous-time programming problems. Subsequently, the structure and contents of these optimality results are utilized as a basis for constructing two parametric and four parameter-free duality models, and for proving weak, strong, and strict converse duality theorems. Furthermore, it is briefly pointed out how similar optimality and duality results can be derived for two special cases of the main problem containing arbitrary norms and square roots of positive semidefinite quadratic forms. These results generalize a number of similar results in the area of continuous-time programming, and subsume a great variety of cognate results originally developed for finite-dimensional nonlinear and fractional programming problems.

Continuous-time multiobjective fractional programming

Both parametric and semiparametric necessary and sufficient proper efficiency principles are established for a class of continuous-time multiobjective fractional programming problems. Based on the forms and contents of these proper efficiency results, two parametric and four semiparametric duality models are constructed and, in each case, weak and strong duality theorems are proved. These proper efficiency and duality results contain, as special cases, similar results for continuous-time programming problems with multiple nonfractional, single fractional, and conventional objective functions. These results improve and generalize a number of existing results in the area of continuous-time programming and, moreover, provide continuous-time analogues of various kindred results previously obtained for certain classes of finitedimensional nonlinear programming problems.

Contiuous time inverse vector optimization problems

In this paper we present a continuous time generalization of nonlinear inverse vector optimization problem. It is shown that finding a solution to inverse vector optimization problem which is efficient with respect to the constraints as well as to the objectives is equivalent to solving a single scalar valued continuous time programming problem. Finally, as a special case, continuous linear inverse vector optimizaton problem is considered.

Generalized sufficiency criteria in continuous-time programming with application to a class of variational-type inequalities

In this paper, we establish a number of sufficiency criteria under generalized ϱ − (η, θ)-invexity conditions for general continuous-time programming problems with nonlinear operator equality and inequality constraints. As an application of these results, we establish some existence criteria for solutions of a class of variational type inequalities.

Optimality conditions and duality for a class of continuous-time programming problems with nonlinear operator equality and inequality constraints

Using a minmax approach, we establish saddle point optimality conditions and Lagrangian duality relations under generalized convexity hypotheses for a class of continuous-time programming problems with nonlinear operator equality and inequality constraints. We also discuss the implications of these results for differentiable problems.

Some remarks concerning duality for continuous-time programming problems

This paper attempts to present a new methodological approach to the study of duality for continuous-time programming, based on the use of the Lebesgue-Stieltjes integral. This approach allows for a unified treatment, including cases when—in more classical approach—either the primal or the dual problem has no feasible or no optimal solution, while the other one has an optimal solution. The continuous-time linear programming problem is discussed in more detail.

On the form of solutions to the linear continuous time programming problem and a conjecture by Tyndall

This paper gives the general form of solutions to the linear continuous time programming problem and shows that the solutions are piecewise smooth.

Post-optimality sensitivity analysis in abstract spaces with applications to continuous-time programming problems

The programming problem under consideration consists in maximizing a concave objective functional, subject to convex operator inequality contraints. The assumptions include the existence of an optimum solution, Frechet differentiability of all operators involved, and the existence of the topological complement of the null space of the Frechet derivative of the constraint operator. It is shown that the rate of change of the optimum value of the objective functional due to the perturbation is measured by the dual. The optimum values of the primal variables are locally approximated as linear functions of the perturbation; the theory of generalized inverse operators is used in the approximation. We give an approximation to the primal variables if the problem is perturbed. The results are specialized for some continuous-time and finite-dimensional cases. Two examples for finite-dimensional problems are given. We apply the theory to the continuous-time linear programming problem and prove some continuity results for the optimal primal and dual objective functionals.

Sufficient optimality criteria in continuous time programming

Sufficient optimality conditions are obtained in the case of continuous time programming problems under the assumptions that (i) particular linear combinations of the components of the constraint function are quasiconvex and objective functional is pseudoconcave “almost everywhere,” (ii) a particular linear combination of constraint function and objective functional is pseudoconcave “almost everywhere”.