Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Propagation by Selective Initialization and Its Application to Numerical Constraint Satisfaction Problems

M. H. van Emden, B. Moa

TL;DR

The paper tackles numerically realistic optimization by solving systems of nonlinear inequalities $g_i(\mathbf{x}) \le 0$ using constraint programming and interval arithmetic, preserving guarantees about containing all solutions and proving infeasibility. It introduces Propagation with Selective Initialization (PSI), a simple modification of the generic propagation algorithm that leverages the structure of composite arithmetic expressions to achieve results equivalent to interval arithmetic with far fewer activations. The authors develop a canonical form for nonlinear inequalities, translate them into Interval CSPs (ICSPs) with a virtual-machine-like DRO execution, and demonstrate how PSI can reproduce interval-evaluation behavior while enabling efficient propagation and parallelism. They also extend the approach to search and large-scale optimization, outlining practical parallelization and future work to handle conjunctions of inequalities with reduced redundant computation.

Abstract

Numerical analysis has no satisfactory method for the more realistic optimization models. However, with constraint programming one can compute a cover for the solution set to arbitrarily close approximation. Because the use of constraint propagation for composite arithmetic expressions is computationally expensive, consistency is computed with interval arithmetic. In this paper we present theorems that support, selective initialization, a simple modification of constraint propagation that allows composite arithmetic expressions to be handled efficiently.

Propagation by Selective Initialization and Its Application to Numerical Constraint Satisfaction Problems

TL;DR

The paper tackles numerically realistic optimization by solving systems of nonlinear inequalities using constraint programming and interval arithmetic, preserving guarantees about containing all solutions and proving infeasibility. It introduces Propagation with Selective Initialization (PSI), a simple modification of the generic propagation algorithm that leverages the structure of composite arithmetic expressions to achieve results equivalent to interval arithmetic with far fewer activations. The authors develop a canonical form for nonlinear inequalities, translate them into Interval CSPs (ICSPs) with a virtual-machine-like DRO execution, and demonstrate how PSI can reproduce interval-evaluation behavior while enabling efficient propagation and parallelism. They also extend the approach to search and large-scale optimization, outlining practical parallelization and future work to handle conjunctions of inequalities with reduced redundant computation.

Abstract

Numerical analysis has no satisfactory method for the more realistic optimization models. However, with constraint programming one can compute a cover for the solution set to arbitrarily close approximation. Because the use of constraint propagation for composite arithmetic expressions is computationally expensive, consistency is computed with interval arithmetic. In this paper we present theorems that support, selective initialization, a simple modification of constraint propagation that allows composite arithmetic expressions to be handled efficiently.

Paper Structure

This paper contains 20 sections, 4 theorems, 8 equations, 4 figures.

Table of Contents

  1. Introduction
  2. A software architecture for optimization problems
  3. Preliminaries
  4. Constraint satisfaction problems
  5. Constraints
  6. Constraint propagation
  7. Intervals
  8. Solving inequalities in interval arithmetic
  9. Interval CSPs
  10. A canonical form for nonlinear numerical inequalities
  11. Translating nonlinear inequalities to ICSPs
  12. The tree form of a formula
  13. Translating a formula to an ICSP
  14. Search
  15. Propagation with selective initialization
  16. ...and 5 more sections

Key Result

lemma 1

Let $t$ be a term that can be evaluated in interval arithmetic. Let the variables of $t$ be $x_1,\ldots,x_n$. Let $y$ be the variable associated with the root of the tree form of $t$. Let $S$ be the ICSP that results from translating $t$, where the domains of $x_1,\ldots,x_n$ are $X_1,\ldots,X_n$ an

Figures (4)

  • Figure 1: A software architecture for optimization problems.
  • Figure 2: A pseudo-code for GPA.
  • Figure 3: A system of non-linear inequalities without multiple occurrences of variables. Instead, the set $\{y_1,\ldots,y_n\}$ is partitioned into equivalence classes $V_1,\ldots,V_p$ where $V_j$ is a subset $\{v_{j,1}, \ldots, v_{j,{n_j}}\}$ of $\{y_1,\ldots,y_n\}$, for $j \in \{1,\ldots, p\}$. An $\mathit{allEq}$ constraint asserts that its arguments are equal.
  • Figure 4: Pseudo-code for Propagation with Selective Initialization (PSI).

Theorems & Definitions (10)

  • lemma 1
  • proof
  • definition 1
  • definition 2
  • theorem 1
  • proof
  • theorem 2
  • proof
  • theorem 3
  • proof