On Robust Saddle-Point Criterion in Optimization Problems with Curvilinear Integral Functionals

Savin Treanţă,Koushik Das

doi:10.3390/math9151790

Abstract

In this paper, we introduce a new class of multi-dimensional robust optimization problems (named (P)) with mixed constraints implying second-order partial differential equations (PDEs) and inequations (PDIs). Moreover, we define an auxiliary (modified) class of robust control problems (named (P)(b¯,c¯)), which is much easier to study, and provide some characterization results of (P) and (P)(b¯,c¯) by using the notions of normal weak robust optimal solution and robust saddle-point associated with a Lagrange functional corresponding to (P)(b¯,c¯). For this aim, we consider path-independent curvilinear integral cost functionals and the notion of convexity associated with a curvilinear integral functional generated by a controlled closed (complete integrable) Lagrange 1-form.

Highlights

Robust Saddle-Point Criterion inAs we all know, partial differential equations (PDEs) and partial differential inequations (PDIs) are essential in modeling and investigating many processes in engineering and science
For example, the research works of Mititelu [1], Treanţă [2,3,4], Mititelu and Treanţă [5], Olteanu and Treanţă [6], Preeti et al [7], and Jayswal et al [8] on the study of some optimization problems with ODE, PDE, or isoperimetric constraints
Since the real-life processes and phenomena often imply uncertainty in initial data, many researchers have turned their attention to optimization issues governed by first- and second-order PDEs, isoperimetric restrictions, stochastic PDEs, uncertain data, or a combination thereof

Summary

Introduction

Partial differential equations (PDEs) and partial differential inequations (PDIs) are essential in modeling and investigating many processes in engineering and science. Since the real-life processes and phenomena often imply uncertainty in initial data, many researchers have turned their attention to optimization issues governed by first- and second-order PDEs, isoperimetric restrictions, stochastic PDEs, uncertain data, or a combination thereof In this context, we mention the following research papers: Wei et al [13], Liu and Yuan [14], Jeyakumar et al [15], Sun et al [16], Preeti et al [7], Lu et al [17], and Treanţă [18]. By taking curvilinear integral objective functionals with mixed (equality and inequality) constraints implying data uncertainty and second-order partial derivatives, we introduce the robust control problems under study.

Preliminaries

Saddle-Point Optimality Criterion

Conclusions and Further Development