Neural network model for mathematical programming problems with complementary constraints

Abstract

In this paper, we propose a a gradient-based neural network model to solve the mathematical programming problems with complementary constraints (MPCC). In order to facilitate tractable optimization, the problem MPCC is transformed via a regularized approach into a relaxed nonlinear optimization problem NLP($β$). After that employing the penalty function and neural network model an estimate of the optimal solution of the problem NLP($β$) is obtained. On the basis of Lyapunov stability theory and LaSalle invariance principle, the equilibrium point of proposed neural network is theoretically proven to be asymptotically stable and capable to generate optimal solution of the problem MPCC. Further, we demonstrate the performance and dynamic behavior of the proposed neural network through various illustrative examples and its effectiveness via theoretical and numerical experiments.

Figures (12)

• Figure 1: Graphical view of the stationary conditions for an index $k\in\mathcal{I}_{00}(w^*).$
• Figure 2: Geometrical view of the $\mathcal{B}_k(w, \beta).$
• Figure 3: The architectural design of the proposed neural network model \ref{['eq:3']}.
• Figure :
• Figure :

Theorems & Definitions (16)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Lemma 2.1
• Theorem 2.1
• Theorem 3.1
• proof
• Theorem 4.1
• proof
• Theorem 4.2