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Convex Mixed-Integer Nonlinear Programs Derived from Generalized Disjunctive Programming using Cones

David E. Bernal Neira, Ignacio E. Grossmann

TL;DR

The reformulations, examples, and computational results that support the claim that taking advantage of conic formulations of convex GDP instead of their nonlinear algebraic descriptions can lead to a more efficient solution to these problems are provided.

Abstract

We propose the formulation of convex Generalized Disjunctive Programming (GDP) problems using conic inequalities leading to conic GDP problems. We then show the reformulation of conic GDPs into Mixed-Integer Conic Programming (MICP) problems through both the big-M and hull reformulations. These reformulations have the advantage that they are representable using the same cones as the original conic GDP. In the case of the hull reformulation, they require no approximation of the perspective function. Moreover, the MICP problems derived can be solved by specialized conic solvers and offer a natural extended formulation amenable to both conic and gradient-based solvers. We present the closed form of several convex functions and their respective perspectives in conic sets, allowing users to formulate their conic GDP problems easily. We finally implement a large set of conic GDP examples and solve them via the scalar nonlinear and conic mixed-integer reformulations. These examples include applications from Process Systems Engineering, Machine learning, and randomly generated instances. Our results show that the conic structure can be exploited to solve these challenging MICP problems more efficiently. Our main contribution is providing the reformulations, examples, and computational results that support the claim that taking advantage of conic formulations of convex GDP instead of their nonlinear algebraic descriptions can lead to a more efficient solution to these problems.

Convex Mixed-Integer Nonlinear Programs Derived from Generalized Disjunctive Programming using Cones

TL;DR

The reformulations, examples, and computational results that support the claim that taking advantage of conic formulations of convex GDP instead of their nonlinear algebraic descriptions can lead to a more efficient solution to these problems are provided.

Abstract

We propose the formulation of convex Generalized Disjunctive Programming (GDP) problems using conic inequalities leading to conic GDP problems. We then show the reformulation of conic GDPs into Mixed-Integer Conic Programming (MICP) problems through both the big-M and hull reformulations. These reformulations have the advantage that they are representable using the same cones as the original conic GDP. In the case of the hull reformulation, they require no approximation of the perspective function. Moreover, the MICP problems derived can be solved by specialized conic solvers and offer a natural extended formulation amenable to both conic and gradient-based solvers. We present the closed form of several convex functions and their respective perspectives in conic sets, allowing users to formulate their conic GDP problems easily. We finally implement a large set of conic GDP examples and solve them via the scalar nonlinear and conic mixed-integer reformulations. These examples include applications from Process Systems Engineering, Machine learning, and randomly generated instances. Our results show that the conic structure can be exploited to solve these challenging MICP problems more efficiently. Our main contribution is providing the reformulations, examples, and computational results that support the claim that taking advantage of conic formulations of convex GDP instead of their nonlinear algebraic descriptions can lead to a more efficient solution to these problems.

Paper Structure

This paper contains 23 sections, 2 theorems, 42 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Contributions
  3. Generalized Disjunctive Programming
  4. Conic Generalized Disjunctive Programming
  5. Computational results
  6. Quadratic problems
  7. Constrained layout problem
  8. k-means clustering
  9. Random examples
  10. Results
  11. Exponential problems
  12. Process Networks
  13. Simultaneous Retrofit and Synthesis problems
  14. Logistic Regression
  15. Random examples
  16. ...and 8 more sections

Key Result

Theorem 1

lodi2019disjunctive Let $\mathcal{P}_i = \left \{ \mathbf{x}\in \mathbb{R}^n: \mathbf{A}_{i}\mathbf{x} \succcurlyeq_{\mathcal{K}_{i}} \mathbf{b}_{i} \right \}$ for $i \in I$, where $\mathbf{A}_{i} \in \mathbb{R}^{m_i \times n}$, $\mathbf{b}_{i} \in \mathbb{R}^{m_i}$, and $\mathcal{K}_{i}$ is a prope Then $\textnormal{conv}(\bigcup_{i \in I} \mathcal{P}_i) \subseteq \textnormal{proj}_{\mathbf{x}}(\

Figures (3)

  • Figure 1: Time (left) and nodes (right) absolute performance profile for quadratic instances using the different GDP reformulations and commercial solvers.
  • Figure 2: Time (left) and Nodes (right) absolute performance profile for exponential instances using the different GDP reformulations and commercial solvers.
  • Figure 3: Time (left) and Solved subproblems (right) absolute performance profile for all instances using the different GDP reformulations and solvers through SBB. For the time profiles we include the best performing commercial solver results for each reformulation.

Theorems & Definitions (4)

  • Theorem 1
  • proof
  • Theorem 2
  • proof