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Solving Chance Constrained Programs via a Penalty based Difference of Convex Approach

Zhiping Li, Nan Jiang, Rujun Jiang

Abstract

We develop two penalty based difference of convex (DC) algorithms for solving chance constrained programs. First, leveraging a rank-based DC decomposition of the chance constraint, we propose a proximal penalty based DC algorithm in the primal space that does not require a feasible initialization. Second, to improve numerical stability in the general nonlinear settings, we derive an equivalent lifted formulation with complementary constraints and show that, after minimizing primal variables, the penalized lifted problem admits a tractable DC structure in the dual space over a simple polyhedron. We then develop a penalty based DC algorithm in the lifted space with a finite termination guarantee. We establish exact penalty and stationarity guarantees under mild constraint qualifications and identify the relationship of the local minimizers between the two formulations. Numerical experiments demonstrate the efficiency and effectiveness of our proposed methods compared with state-of-the-art benchmarks.

Solving Chance Constrained Programs via a Penalty based Difference of Convex Approach

Abstract

We develop two penalty based difference of convex (DC) algorithms for solving chance constrained programs. First, leveraging a rank-based DC decomposition of the chance constraint, we propose a proximal penalty based DC algorithm in the primal space that does not require a feasible initialization. Second, to improve numerical stability in the general nonlinear settings, we derive an equivalent lifted formulation with complementary constraints and show that, after minimizing primal variables, the penalized lifted problem admits a tractable DC structure in the dual space over a simple polyhedron. We then develop a penalty based DC algorithm in the lifted space with a finite termination guarantee. We establish exact penalty and stationarity guarantees under mild constraint qualifications and identify the relationship of the local minimizers between the two formulations. Numerical experiments demonstrate the efficiency and effectiveness of our proposed methods compared with state-of-the-art benchmarks.
Paper Structure (19 sections, 18 theorems, 107 equations, 3 tables, 2 algorithms)

This paper contains 19 sections, 18 theorems, 107 equations, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Relevant Literature
  3. Summary of Contributions
  4. Notations and Preliminaries
  5. Algorithm Frameworks
  6. Algorithm in the Primal Space
  7. Equivalent Formulation in the Lifted Space
  8. Algorithm in the Lifted Space
  9. Exact Penalty Frameworks
  10. Global Exact Penalty
  11. Local Exact Penalty of the Primal Formulation
  12. Equivalence between (Strongly) Stationary Points of the Lifted Formulation
  13. Identifying Spurious Local Minimizers
  14. Numerical Experiments
  15. A VaR-constrained Portfolio Optimization Problem
  16. ...and 4 more sections

Key Result

Proposition 1.1

Suppose GMFCQ eq:prelim-gmfcq holds at $\bar{\boldsymbol{x}}$. Define $\mathcal{G}(\boldsymbol{x}):=g(\boldsymbol{x})+\mathbb{R}_+$ for $\boldsymbol{x}\in\mathcal{X}$. Then $\mathcal{F}$ is metrically regular at $\bar{\boldsymbol{x}}$. Consequently, there exist $\kappa>0$ and $\varepsilon>0$ such th holds.

Theorems & Definitions (31)

  • Proposition 1.1
  • Theorem 1.2: Local exact penalty
  • Theorem 1.3: Global exact penalty
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • Proposition 2.5
  • proof
  • ...and 21 more