Normalized multiparametric disaggregation: an efficient relaxation for mixed-integer bilinear problems

Abstract

A key element for the global optimization of non-convex mixed-integer bilinear problems is the computation of a tight lower bound for the objective function being minimized. Multiparametric disaggregation is a technique for generating a mixed-integer linear relaxation of a bilinear problem that works by discretizing the domain of one of the variables in every bilinear term according to a numeric representation system. This can be done up to a certain accuracy level that can be different for each discretized variable so as to adjust the number of significant digits to their range of values and give all variables the same importance. We now propose a normalized formulation (NMDT) that achieves the same goal using a common setting for all variables, which is equivalent to the number of uniform partitions in a closely related, piecewise McCormick (PCM) approach. Through the solution of several benchmark problems from the literature involving four distinct problem classes, we show that the computational performance of NMDT is already better than PCM for ten partitions, with the difference rising quickly due to the logarithmic versus linear growth in the number of binary variables with the number of partitions. The results also show that a global optimization solver based on the proposed relaxation compares favorably with commercial solvers BARON and GloMIQO.

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Abbreviations

\(a_{ijq}\) :

Scalar multiplying bilinear term \(x_i x_j \) in constraint q

\(B_q\) :

Matrix with coefficients for variables \({\varvec{x}}\) in constraint q

\(C_q\) :

Matrix with coefficients for variables \({\varvec{y}}\) in constraint q

\(d_q\) :

Constant term in constraint q

\(f_0\) :

Optimal value of objective function for problem (P)

\(f_0^{\prime }\) :

Optimal value of objective function for problem (P’) that is equivalent to (P)

\(f_0^R\) :

Optimal value of objective function for problem (PR), lower bound for (P)

\(f_0^*\) :

Upper bound for problem (P)

\(f_q \left( {{\varvec{x}},{\varvec{y}}} \right) \) :

Function of continuous variables x and binary variables y defining constraint q

k :

Digit in decimal numerical representation system, \(\in \left\{ {0,\ldots ,9} \right\} \)

l :

Position in decimal numerical representation system, \(\in \left\{ {p,\ldots ,-1} \right\} \)

n :

Partition in piecewise McCormick relaxation, \(\in \left\{ {1,\ldots ,N} \right\} \)

N :

Number of partitions specified for piecewise McCormick relaxation

p :

Parameter defining the accuracy level of discretized variables, \(\in {\mathbb {Z}}^{-}\)

\({\varvec{x}}^{L}\) :

Vector of lower bounds of continuous variables x

\({\varvec{x}}^{U}\) :

Vector of upper bounds of continuous variables x

\(x_i\) :

Variable of bilinear term \(x_i x_j \) to be disaggregated

\({\hat{x}}_{ijkl}\) :

Disaggregated variable from linearization of \(x_i z_{jkl} \)

\({\hat{x}}_{ijn}\) :

Disaggregated variable in piecewise McCormick relaxation

\({\hat{x}}_{jn}\) :

Disaggregated variable in piecewise McCormick relaxation

\(x_j\) :

Variable of bilinear term \(x_i x_j \) to be discretized

\(w_{ij}\) :

Variable replacing bilinear term \(x_i x_j \)

\(z_{jkl}\) :

Binary variable assigning to \(\lambda _j \) digit k to position l

\(z_{jn}\) :

Binary variable assigning partition n to variable \(x_j \)

\(\varepsilon \) :

Targeted relative optimality tolerance \(\lambda _j=\) Discretized variable linked to original variable \(x_j \), \(\in \left[ {0,1} \right] \)

\(\lambda _{jl}\) :

Value of \(\lambda _j \) in position l of discretized representation, \(\in \left\{ {0,10^{l},\ldots ,9\cdot 10^{l}} \right\} \)

\(\nu _{ij}\) :

Variable replacing bilinear term \(x_i \lambda _j \)

\(\tau \) :

Parameter for comparing performance of a solver with respect to its competitors

\(\Delta \lambda _j\) :

Slack variable ensuring continuous domain for \(\lambda _j , \in \left[ {0,10^{p}} \right] \)

\(\Delta \nu _{ij}\) :

Variable replacing bilinear term \(x_i \cdot \Delta \lambda _j \)

\(\left( {{\varvec{x}}^{R},{\varvec{y}}^{R}} \right) \) :

Optimal solution for problem (PR)

\(\left( {{\varvec{x}}^{*},{\varvec{y}}^{*}} \right) \) :

Best-known solution for problem (P)

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Acknowledgments

Financial support from Fundação para a Ciência e Tecnologia (FCT) through the Investigador FCT 2013 program.

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Correspondence to Pedro M. Castro.

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Castro, P.M. Normalized multiparametric disaggregation: an efficient relaxation for mixed-integer bilinear problems. J Glob Optim 64, 765–784 (2016). https://doi.org/10.1007/s10898-015-0342-z

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Keywords

  • Mixed-integer nonlinear programming
  • Quadratic optimization
  • Disjunctive programming
  • Algorithm
  • Process networks