Multi-dimensional Variational Control Problem with Data Uncertainty in Constraint Functionals

Abstract

In the last few years, the applicability of the penalty function method, initiated by Zangwill [1] for the constrained optimization problem, has grown significantly. The penalty function approach transforms the constrained optimization problem into an unconstrained optimization problem and preserves the optimality of the original one. In this way, the solution sets of the unconstrained optimization problems ideally converge to the solution sets of the constrained optimization problems. The idea behind the penalty function approach (the convergence of the solution sets for constrained optimization problem and its associated unconstrained optimization problem) encourages the researchers to establish the equivalence between the solution set of constrained and unconstrained problems under suitable assumptions for different kinds of optimization problems. Antczak [2] used an exact $l_{1}$ penalty function method in convex nondifferentiable multi-objective optimization problem and established the equivalence between the solution set of the original problem and its associated penalized problem. Also, Alvarez [3], Antczak [4] and Liu and Feng [5] explored the exponential penalty function method for multi-objective optimization problem and established the relationships between the constrained and unconstrained optimization problems. On the other hand, Li et al. [6] used the penalty function method to solve the continuous inequality constrained optimal control problem. Thereafter, Jayswal and Preeti [7] extended the applicability of the penalty function method for the multi-dimensional optimization problem. Moreover, Jayswal et al. [8] explored the same for uncertain optimization problem under convexity assumptions.