Multitime multiobjective variational problems via η-approximation method

Abstract

The present article is devoted to multitime multiobjective variational problems via ?-approximation method. In this method, an ?-approximation approach is applied to the considered problem, and a new problem is constructed, called as ?-approximated multitime multiobjective variational problem that contains the change in objective and both constraints functions. The equivalence between an efficient (Pareto optimal) solution to the main multitime multiobjective variational problem is derived along with its associated ?-approximated problem under invexity defined for a multi- time functional. Furthermore, we have also discussed the saddle-point criteria for the problem considered and its associated ?-approximated problems via generalized invexity assumptions.

Solving invex multitime control problems with first‐order PDE constraints via the absolute value exact penalty method

“…We now use the absolute value exact penalty function method for solving (MCP1). Then, we construct its associated penalized control problem (MCP1) as follows P((x(t), u(t)), 𝜍(t)) = ∫ Ω 0, 1…”

“…This is a consequence of various applications of multitime control optimization problems with first‐order PDE constraints in many areas of human activity in which modern man lives and works, for example, in economics, mechanics, engineering, physics, and others. Therefore, the aforementioned control optimization problems have been intensively studied in the literature in the last few years (see, e.g., References 1–9, and others).…”

“…Further, by the PDE constraints, it follows that u(t) = tan , 𝛿 2 ), where 𝛿 1 , 𝛿 2 ∈ R are constant,such that 𝛿 1 + 𝛿 2 = −1 and 𝜇 1 (t) = 0, 𝜇 2 (t) = 0, ∀t ∈ Ω 0, 1 2 .Hence, the necessary optimality conditions (2)-(5) are satisfied with Lagrange multipliers 𝜇 1 (t) = 0, 𝜇 2 (t) = 0, 𝜗 = 0, t 2 = 0 is an optimal solution in (MCP1). We have F(x(⋅), u(⋅)) = ∫ Ω 0,1…”