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Dynamical System Approach for Optimal Control Problems with Equilibrium Constraints Using Gap-Constraint-Based Reformulation

Kangyu Lin, Toshiyuki Ohtsuka

TL;DR

The paper tackles optimal control problems with equilibrium constraints (OCPEC) governed by differential variational inequalities (DVI), where direct discretize-then-optimize methods struggle due to nonsmooth dynamics and constraint qualification violations. It introduces gap-constraint-based reformulations via Auchmuty gap functions to smooth the DVI into a small, semismoothly differentiable set of constraints, enabling multiplier-free NLP subproblems. A fictitious-time semismooth Newton flow is proposed to solve a sequence of these smoothed subproblems through a continuation in the smoothing parameter, with proven exponential convergence under standard assumptions. A numerical example on a linear complementarity system demonstrates fast convergence and competitive computation times, validating the method’s potential for efficient OCPEC solving in practice.

Abstract

This study focuses on using direct methods (first-discretize-then-optimize) to solve optimal control problems for a class of nonsmooth dynamical systems governed by differential variational inequalities (DVI), called optimal control problems with equilibrium constraints (OCPEC). In the discretization step, we propose a class of novel approaches to smooth the DVI. The generated smoothing approximations of DVI, referred to as gap-constraint-based reformulations, have computational advantages owing to their concise and semismoothly differentiable constraint system. In the optimization step, we propose an efficient dynamical system approach to solve the discretized OCPEC, where a sequence of its smoothing approximations is solved approximately. This system approach involves a semismooth Newton flow, thereby achieving fast local exponential convergence. We confirm the effectiveness of our method using a numerical example.

Dynamical System Approach for Optimal Control Problems with Equilibrium Constraints Using Gap-Constraint-Based Reformulation

TL;DR

The paper tackles optimal control problems with equilibrium constraints (OCPEC) governed by differential variational inequalities (DVI), where direct discretize-then-optimize methods struggle due to nonsmooth dynamics and constraint qualification violations. It introduces gap-constraint-based reformulations via Auchmuty gap functions to smooth the DVI into a small, semismoothly differentiable set of constraints, enabling multiplier-free NLP subproblems. A fictitious-time semismooth Newton flow is proposed to solve a sequence of these smoothed subproblems through a continuation in the smoothing parameter, with proven exponential convergence under standard assumptions. A numerical example on a linear complementarity system demonstrates fast convergence and competitive computation times, validating the method’s potential for efficient OCPEC solving in practice.

Abstract

This study focuses on using direct methods (first-discretize-then-optimize) to solve optimal control problems for a class of nonsmooth dynamical systems governed by differential variational inequalities (DVI), called optimal control problems with equilibrium constraints (OCPEC). In the discretization step, we propose a class of novel approaches to smooth the DVI. The generated smoothing approximations of DVI, referred to as gap-constraint-based reformulations, have computational advantages owing to their concise and semismoothly differentiable constraint system. In the optimization step, we propose an efficient dynamical system approach to solve the discretized OCPEC, where a sequence of its smoothing approximations is solved approximately. This system approach involves a semismooth Newton flow, thereby achieving fast local exponential convergence. We confirm the effectiveness of our method using a numerical example.

Paper Structure

This paper contains 19 sections, 10 theorems, 38 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background, motivation and related works
  3. Contribution and outline
  4. Preliminaries and problem formulation
  5. Notation, nonsmooth analysis and variational inequalities
  6. Optimal control problem with equilibrium constraints
  7. Proposed approaches to smoothing the DVI
  8. Gap-constraint-based reformulations
  9. Computation considerations, constraint regularity, and geometric interpretation
  10. Dynamical system approach to solve OCPEC
  11. Problem setting and assumptions
  12. Fictitious-time semismooth Newton flow dynamical system
  13. Convergence analysis
  14. Numerical experiment
  15. Conclusion
  16. ...and 4 more sections

Key Result

Proposition 1

The following three statements are valid for gap functions $\varphi^{c}_{Au}(\lambda, \eta)$ and $\varphi^{ab}_{Au}(\lambda, \eta)$:

Figures (3)

  • Figure 1: Geometric interpretation of the gap-constraint-based reformulations
  • Figure 2: History of KKT residual and computation time (Proposed method).
  • Figure 3: History of KKT error and computation time (Classical method).

Theorems & Definitions (18)

  • Definition 1
  • Proposition 1
  • Remark 1
  • Proposition 2
  • Theorem 1
  • Example 1
  • Remark 2
  • Lemma 1
  • Theorem 2
  • Remark 3
  • ...and 8 more