Semidefinite Multiobjective Mathematical Programming Problems with Vanishing Constraints Using Convexificators

Kin Keung Lai,Mohd Hassan,Sanjeev Kumar Singh,Shashi Kant Mishra,Jitendra Kumar Maurya

doi:10.3390/fractalfract6010003

Kin Keung Lai, Mohd Hassan + Show 3 more

Open Access

https://doi.org/10.3390/fractalfract6010003

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Abstract

In this paper, we establish Fritz John stationary conditions for nonsmooth, nonlinear, semidefinite, multiobjective programs with vanishing constraints in terms of convexificator and introduce generalized Cottle type and generalized Guignard type constraints qualification to achieve strong S—stationary conditions from Fritz John stationary conditions. Further, we establish strong S—stationary necessary and sufficient conditions, independently from Fritz John conditions. The optimality results for multiobjective semidefinite optimization problem in this paper is related to two recent articles by Treanta in 2021. Treanta in 2021 discussed duality theorems for special class of quasiinvex multiobjective optimization problems for interval-valued components. The study in our article can also be seen and extended for the interval-valued optimization motivated by Treanta (2021). Some examples are provided to validate our established results.

Highlights

Nonlinear semidefinite programming problems (SDP) include several classes of optimization problems, such as linear programming, quadratic programming, second order cone programming [1], and semidefinite programming [2]
Discussed simple extensions of constant rank-type constraint qualifications to semidefinite programming, which are based on the Approximate Karush–Kuhn–Tucker necessary optimality condition and on the application of the reduction approach
We propose some index sets to show sufficient optimality conditions for SMMPVC:

Summary

Introduction

Nonlinear semidefinite programming problems (SDP) include several classes of optimization problems, such as linear programming, quadratic programming, second order cone programming [1], and semidefinite programming [2]. The nonlinear semidefinite programming problem has broad applications in system control [3], truss topology optimization [4], and other several fields. Golestani and Nobakhtian [11] proposed the generalized Abadie constraint qualification ( GACQ) and established necessary and sufficient optimality conditions for nonlinear semidefinite programming problems using convexificators. Discussed simple extensions of constant rank-type constraint qualifications to semidefinite programming, which are based on the Approximate Karush–Kuhn–Tucker necessary optimality condition and on the application of the reduction approach. Motivated by the above mentioned work, we propose some new constraints qualification to establish necessary and sufficient type optimality conditions for nonsmooth, nonlinear, semidefinite, multiobjective mathematical programs with vanishing constraints.

Preliminaries

Optimality Conditions

Conclusions and Future Remarks