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Warm-starting outer approximation for parametrized convex MINLP

Erik Tamm, Gabriele Eichfelder, Jan Kronqvist

TL;DR

The paper tackles efficient resolution of parameterized sequences of convex MINLPs using warm-started outer approximation (OA). It develops a theoretical basis showing that, under suitable conditions, reusing or tightening the polyhedral outer approximation can yield convergence in a single OA iteration, and proposes three practical warm-starting rules that exploit problem similarity. Empirical results on biobjective problems, sparse regression, and MPC demonstrate that cut-tightening and point-based strategies substantially reduce the number of MILP subproblems and total solve time, particularly when the integer structure is stable across parameter values. The findings highlight significant practical potential for accelerating parameterized MINLP workflows and point to future work on cut management and integration with established solvers.

Abstract

We address the challenge of efficiently solving parameterized sequences of convex Mixed-Integer Nonlinear Programming (MINLP) problems through warm-starting techniques. We focus on an outer approximation (OA) approach, for which we develop the theoretical foundation and present two warm-starting techniques for solving sequences of convex MINLPs. These types of problem sequences arise in several important applications, such as, multiobjective MINLPs using scalarization techniques, sparse linear regression, hybrid model predictive control, or simply in analyzing the impact of certain problem parameters. The main contribution of this paper is the mathematical analysis of the proposed warm-starting framework for OA-based algorithms, which shows that a simple adaptation of the polyhedral outer approximation from one problem to the next can greatly improve the computational performance. We prove that, under some conditions, one of the proposed warm-starting techniques result in only one OA iteration to find an optimal solution and verify optimality. Numerical results demonstrate noticeable performance improvements compared to two common initialization approaches, and show that the warm-starting can also in practice result in a single iteration to converge for several problems in the sequences. Our methods are especially effective for problems where consecutive problems in the sequence are similar, and where the integer part of the optimal solutions are constant for several problems in the sequence. The results show that it is possible, both in theory and practice, to perform warm-starting to significantly enhance the computational efficiency of solving parameterized convex MINLPs.

Warm-starting outer approximation for parametrized convex MINLP

TL;DR

The paper tackles efficient resolution of parameterized sequences of convex MINLPs using warm-started outer approximation (OA). It develops a theoretical basis showing that, under suitable conditions, reusing or tightening the polyhedral outer approximation can yield convergence in a single OA iteration, and proposes three practical warm-starting rules that exploit problem similarity. Empirical results on biobjective problems, sparse regression, and MPC demonstrate that cut-tightening and point-based strategies substantially reduce the number of MILP subproblems and total solve time, particularly when the integer structure is stable across parameter values. The findings highlight significant practical potential for accelerating parameterized MINLP workflows and point to future work on cut management and integration with established solvers.

Abstract

We address the challenge of efficiently solving parameterized sequences of convex Mixed-Integer Nonlinear Programming (MINLP) problems through warm-starting techniques. We focus on an outer approximation (OA) approach, for which we develop the theoretical foundation and present two warm-starting techniques for solving sequences of convex MINLPs. These types of problem sequences arise in several important applications, such as, multiobjective MINLPs using scalarization techniques, sparse linear regression, hybrid model predictive control, or simply in analyzing the impact of certain problem parameters. The main contribution of this paper is the mathematical analysis of the proposed warm-starting framework for OA-based algorithms, which shows that a simple adaptation of the polyhedral outer approximation from one problem to the next can greatly improve the computational performance. We prove that, under some conditions, one of the proposed warm-starting techniques result in only one OA iteration to find an optimal solution and verify optimality. Numerical results demonstrate noticeable performance improvements compared to two common initialization approaches, and show that the warm-starting can also in practice result in a single iteration to converge for several problems in the sequences. Our methods are especially effective for problems where consecutive problems in the sequence are similar, and where the integer part of the optimal solutions are constant for several problems in the sequence. The results show that it is possible, both in theory and practice, to perform warm-starting to significantly enhance the computational efficiency of solving parameterized convex MINLPs.

Paper Structure

This paper contains 25 sections, 6 theorems, 43 equations, 7 figures, 1 table, 6 algorithms.

Table of Contents

  1. Introduction
  2. Problem formulation and background
  3. Notation
  4. Parameterized convex MINLP
  5. Basic concepts of outer approximation
  6. Basic concepts of multiobjective optimization
  7. Outer Approximation Algorithm
  8. Illustrative example for warm-starting
  9. Theoretical justifications
  10. Properties
  11. Warm-starting algorithm
  12. Baseline: Relaxation
  13. Initialization rule 1: Restart for every parameter value
  14. Initialization rule 2: Cut tightening
  15. Initialization rule 3: Point based
  16. ...and 10 more sections

Key Result

Proposition 3

Let $\mathcal{S} = \{x \in \mathbb{R}^n \mid f(x) \leq 0\}$ for some convex and continuously differentiable function $f\colon\mathbb{R}^n \rightarrow \mathbb{R}$. A linear inequality that separates $\hat{x} \notin \mathcal{S}$ from $\mathcal{S}$ is given by This linear inequality is called a gradient cut.

Figures (7)

  • Figure 1: Illustration of Example \ref{['ex:warm-starting']}. The black line and the black ellipse correspond to the constraints. The green lines correspond to the feasible set. The blue line is the linearization at the optimal solution. The red line is the linearization at the optimal solution of the continuous relaxation of \ref{['eq:worst-case-warm-starting']}.
  • Figure 2: Illustration of Example \ref{['ex:int_jump']}. The black lines and curves correspond to the constraints. The green lines correspond to the feasible set. The blue lines correspond to the linearizations.
  • Figure 3: Results for solving \ref{['eq:ti4']}.
  • Figure 4: Results for solving \ref{['eq:ti14']}.
  • Figure 5: Results for solving \ref{['eq:slr']} by varying $\lambda$.
  • ...and 2 more figures

Theorems & Definitions (22)

  • Definition 1
  • Definition 2
  • Proposition 3
  • Definition 4
  • Definition 5
  • Example 6
  • Definition 7
  • Proposition 8
  • proof
  • Proposition 9
  • ...and 12 more