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An ODE approach to multiple choice polynomial programming

Sihong Shao, Yishan Wu

TL;DR

This work introduces an ODE-based continuous-time algorithm for multiple choice polynomial programming (MCPP), modeling a continuous-time Markov process on the feasible region and using a Boltzmann-type equilibrium to approximate discrete optima. It proves that, for small temperature $T$, the ODE equilibria correspond to local MCPP optima, and provides practical numerical strategies (initialization, adaptive stepping, rounding) to compute these equilibria efficiently. The approach is validated on NP-hard problems such as MAX-$k$-CUT and the star discrepancy, achieving solution quality comparable to state-of-the-art heuristics (MOH and TA_improved) and often surpassing direct MILP solvers like Gurobi within similar runtimes. Additionally, the paper marks the first continuous framework for approximating star discrepancy and outlines avenues to extend the method to broader mixed-integer programming settings and to leverage polyhedral structure for acceleration.

Abstract

We propose an ODE approach to solving multiple choice polynomial programming (MCPP) after assuming that the optimum point can be approximated by the expected value of so-called thermal equilibrium as usually did in simulated annealing. The explicit form of the feasible region and the affine property of the objective function are both fully exploited in transforming the MCPP problem into the ODE system. We also show theoretically that a local optimum of the former can be obtained from an equilibrium point of the latter. Numerical experiments on two typical combinatorial problems, MAX-$k$-CUT and the calculation of star discrepancy, demonstrate the validity of our ODE approach, and the resulting approximate solutions are of comparable quality to those obtained by the state-of-the-art heuristic algorithms but with much less cost. This paper also serves as the first attempt to use a continuous algorithm for approximating the star discrepancy.

An ODE approach to multiple choice polynomial programming

TL;DR

This work introduces an ODE-based continuous-time algorithm for multiple choice polynomial programming (MCPP), modeling a continuous-time Markov process on the feasible region and using a Boltzmann-type equilibrium to approximate discrete optima. It proves that, for small temperature , the ODE equilibria correspond to local MCPP optima, and provides practical numerical strategies (initialization, adaptive stepping, rounding) to compute these equilibria efficiently. The approach is validated on NP-hard problems such as MAX--CUT and the star discrepancy, achieving solution quality comparable to state-of-the-art heuristics (MOH and TA_improved) and often surpassing direct MILP solvers like Gurobi within similar runtimes. Additionally, the paper marks the first continuous framework for approximating star discrepancy and outlines avenues to extend the method to broader mixed-integer programming settings and to leverage polyhedral structure for acceleration.

Abstract

We propose an ODE approach to solving multiple choice polynomial programming (MCPP) after assuming that the optimum point can be approximated by the expected value of so-called thermal equilibrium as usually did in simulated annealing. The explicit form of the feasible region and the affine property of the objective function are both fully exploited in transforming the MCPP problem into the ODE system. We also show theoretically that a local optimum of the former can be obtained from an equilibrium point of the latter. Numerical experiments on two typical combinatorial problems, MAX--CUT and the calculation of star discrepancy, demonstrate the validity of our ODE approach, and the resulting approximate solutions are of comparable quality to those obtained by the state-of-the-art heuristic algorithms but with much less cost. This paper also serves as the first attempt to use a continuous algorithm for approximating the star discrepancy.
Paper Structure (11 sections, 3 theorems, 59 equations, 2 figures, 7 tables)

This paper contains 11 sections, 3 theorems, 59 equations, 2 figures, 7 tables.

Table of Contents

  1. Introduction
  2. The ODE approach: A continuous version of simulated annealing
  3. Numerical analysis
  4. Implementation details
  5. Choice of the initial data
  6. Variable time step
  7. Rounding procedure
  8. Cost analysis
  9. Application to MAX-$k$-CUT
  10. Application to approximating the star discrepancy
  11. Conclusion and discussion

Key Result

Proposition 3.2

The $n$-dimensional vector $\hat{y}$ defined in Eq. eq:equ_inf is a local optimum of MCPP eq:mcpp0 in $\bar{\mathcal{X}}$.

Figures (2)

  • Figure 1: Quality check: Histogram of the ratios between the cut values of the ODE approach and the reference values produced by MOH for all $4\times 71 \times 100$ trials of MAX-$k$-CUT.
  • Figure 2: Quality check: Histogram of the ratios between the discrepancy values of the ODE approach and the reference values produced by TA_improved for all $36 \times 100$ trials.

Theorems & Definitions (9)

  • Example 2.1
  • Definition 3.1: local optimality
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Remark 3.4
  • Proposition 6.1
  • proof