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A successive difference-of-convex method for a class of two-stage nonconvex nonsmooth stochastic conic program via SVI

Chao Zhang, Di Wang

TL;DR

This paper defines a KKT point of the problem, shows that it is a necessary optimality condition under mild conditions, and transforms it to an equivalent nonmonotone nonsmooth two-stage stochastic variational inequality (SVI), and proposes a successive difference-of-convex (SDC) method.

Abstract

We consider a class of two-stage nonconvex nonsmooth stochastic conic program, where the objective functions in both stages can contain nonsmooth terms that are functions with easily computed proximal mappings, further composed with affine mappings. This kind of problem is capable of modeling various applications. Solving these problems, however, can be challenging due to the two-stage structure with possibly large number of scenarios, the nonconvex, nonsmooth and even non-Lipschitz discontinuous terms, as well as the conic constraints. In this paper, we define a KKT point of the problem, show that it is a necessary optimality condition under mild conditions, and transform it to an equivalent nonmonotone nonsmooth two-stage stochastic variational inequality (SVI). We then propose a successive difference-of-convex (SDC) method by making use of Moreau envelope to solve it, the subproblems of which are approximately solved by the progressive hedging method for solving maximal monotone two-stage SVI. We show the rigorous convergence of our method under suitable assumptions. An extension of Markowitz's mean-variance model is provided as an application and numerical results on it demonstrate the effectiveness of the model and the SDC method.

A successive difference-of-convex method for a class of two-stage nonconvex nonsmooth stochastic conic program via SVI

TL;DR

This paper defines a KKT point of the problem, shows that it is a necessary optimality condition under mild conditions, and transforms it to an equivalent nonmonotone nonsmooth two-stage stochastic variational inequality (SVI), and proposes a successive difference-of-convex (SDC) method.

Abstract

We consider a class of two-stage nonconvex nonsmooth stochastic conic program, where the objective functions in both stages can contain nonsmooth terms that are functions with easily computed proximal mappings, further composed with affine mappings. This kind of problem is capable of modeling various applications. Solving these problems, however, can be challenging due to the two-stage structure with possibly large number of scenarios, the nonconvex, nonsmooth and even non-Lipschitz discontinuous terms, as well as the conic constraints. In this paper, we define a KKT point of the problem, show that it is a necessary optimality condition under mild conditions, and transform it to an equivalent nonmonotone nonsmooth two-stage stochastic variational inequality (SVI). We then propose a successive difference-of-convex (SDC) method by making use of Moreau envelope to solve it, the subproblems of which are approximately solved by the progressive hedging method for solving maximal monotone two-stage SVI. We show the rigorous convergence of our method under suitable assumptions. An extension of Markowitz's mean-variance model is provided as an application and numerical results on it demonstrate the effectiveness of the model and the SDC method.
Paper Structure (10 sections, 12 theorems, 129 equations, 3 figures, 3 tables, 2 algorithms)

This paper contains 10 sections, 12 theorems, 129 equations, 3 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Optimality condition for T-NNS-SCP
  4. Solution method for (\ref{['new_prob']})
  5. Successive difference-of-convex method
  6. Theoretical guarantee for global convergence
  7. Useful observations and extensions
  8. Nonconvex nonsmooth two-stage extension of Markowitz's mean-variance model
  9. Numerical experiments
  10. Conclusions

Key Result

Lemma 1

Let $P_0$ be a real valued closed function with $\inf P_0 > -\infty$. Suppose that $w^t \rightarrow w^*, \rho_t \downarrow 0$ and pick any $\zeta^t \in \operatorname{prox}_{\rho_t P_0}\left(w^t\right)$ for each $t$. Then it holds that $\zeta^t \rightarrow w^*$.

Figures (3)

  • Figure 1: Overall framework of the proposed SDC-PHM and its theoretical justification
  • Figure 2: Objective value (blue) and Cardinality (red) versus CPU time for Models A–D under $K=5000$
  • Figure 3: Feasibility Error versus CPU time for Models A–D under $K=5000$

Theorems & Definitions (25)

  • Lemma 1
  • Lemma 2
  • Proposition 1
  • proof
  • Definition 1
  • Lemma 3
  • proof
  • Definition 2: KKT Point
  • Theorem 2
  • proof
  • ...and 15 more