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A Framework for Globally Optimizing Mixed-Integer Signomial Programs

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Abstract

Mixed-integer signomial optimization problems have broad applicability in engineering. Extending the Global Mixed-Integer Quadratic Optimizer, GloMIQO (Misener, Floudas in J. Glob. Optim., 2012. doi:10.1007/s10898-012-9874-7), this manuscript documents a computational framework for deterministically addressing mixed-integer signomial optimization problems to ε-global optimality. This framework generalizes the GloMIQO strategies of (1) reformulating user input, (2) detecting special mathematical structure, and (3) globally optimizing the mixed-integer nonconvex program. Novel contributions of this paper include: flattening an expression tree towards term-based data structures; introducing additional nonconvex terms to interlink expressions; integrating a dynamic implementation of the reformulation-linearization technique into the branch-and-cut tree; designing term-based underestimators that specialize relaxation strategies according to variable bounds in the current tree node. Computational results are presented along with comparison of the computational framework to several state-of-the-art solvers.

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Acknowledgements

C.A.F. is thankful for support from the National Science Foundation (CBET–0827907). This material is based upon work supported by the National Science Foundation Graduate Research Fellowship to R.M. under Grant No. DGE-0646086 and a Royal Academy of Engineering Research Fellowship to R.M.

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  1. Ruth Misener

    Present address: Centre for Process Systems Engineering, Imperial College London, London, UK

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  1. Department of Chemical and Biological Engineering, Princeton University, Princeton, NJ, 08544-5263, USA

    Ruth Misener & Christodoulos A. Floudas

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Correspondence to Christodoulos A. Floudas.

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Appendices

Appendix A: Reformulation Linearization Technique (RLT) Equations

The equations in this section are presented in factored form for clarity, but the MISO framework addresses them in expanded form. The RLT equations are updated throughout the branch-and-cut tree; any upper bound on a variable or a nonlinear term represents the upper bound at the current tree node.

Equation/Variable products may generate as many as four RLT equations (an equation with only one inequality bound produces two RLT equations; equality equations produce exactly one RLT equation):

$$ \begin{array} {l} \displaystyle\phantom{-1 \cdot {}} \Biggl( \sum_{s = 1}^{S_m} c_{m, \, s} \cdot f_{m, \, s}( x) + x^T \cdot Q_m \cdot x + a_m \cdot x - b_m^{\mathrm{U}} \Biggr) \cdot \bigl( x_i - x_i^U \bigr) \geq 0 \\ \displaystyle-1 \cdot \Biggl( \sum _{s = 1}^{S_m} c_{m, \, s} \cdot f_{m, \, s}(x) + x^T \cdot Q_m \cdot x + a_m \cdot x - b_m^{\mathrm{U}} \Biggr) \cdot \bigl( x_i - x_i^L \bigr) \geq 0 \\ \displaystyle-1 \cdot \Biggl( \sum _{s = 1}^{S_m} c_{m, \, s} \cdot f_{m, \, s}(x) + x^T \cdot Q_m \cdot x + a_m \cdot x - b_m^{\mathrm{L}} \Biggr) \cdot \bigl( x_i - x_i^U \bigr) \geq 0 \\ \displaystyle\phantom{-1 \cdot {}} \Biggl( \sum _{s = 1}^{S_m} c_{m, \, s} \cdot f_{m, \, s}(x) + x^T \cdot Q_m \cdot x + a_m \cdot x - b_m^{\mathrm{L}} \Biggr) \cdot \bigl( x_i - x_i^L \bigr) \geq 0 \\ \quad \forall \; m \in \{ 1, \, \ldots, \, M \}; \; i \in C \times B \times I \end{array} $$

Nonlinear Term/Variable products introduce four RLT equations; s∈{1,…,S} denotes a list of the quadratic, bilinear, and signomial terms:

$$ \begin{array} {ll} \phantom{-1 \cdot{}} \bigl( f_{s}(x) - f_{s}(x)^{\mathrm {UP}} \bigr) \cdot \bigl( x_i - x_i^U \bigr) \geq0 & \quad\forall s \in\{ 1, \ldots, S \}; i \in C \times B \times I \\ -1 \cdot \bigl( f_{s}(x) - f_{s}( x)^{\mathrm{UP}} \bigr) \cdot \bigl( x_i - x_i^L \bigr) \geq0 & \quad\forall s \in\{ 1, \ldots, S \}; i \in C \times B \times I \\ -1 \cdot \bigl( f_{s}(x) - f_{s}( x)^{\mathrm{LO}} \bigr) \cdot \bigl( x_i - x_i^U \bigr) \geq0 & \quad\forall s \in\{ 1, \ldots, S \}; i \in C \times B \times I \\ \phantom{-1 \cdot{}} \bigl( f_{s}(x) - f_{s}(x)^{\mathrm {LO}} \bigr) \cdot \bigl( x_i - x_i^L \bigr) \geq0 & \quad\forall s \in\{ 1, \ldots, S \}; i \in C \times B \times I \end{array} $$

Equation/Nonlinear Term products may generate up to four RLT equations; s∈{1,…,S} represents a list of the quadratic, bilinear, and signomial terms:

$$ \begin{array} {l} \displaystyle\phantom{-1 \cdot{}} \Biggl( \sum _{s = 1}^{S_m} c_{m, s} \cdot f_{m, s}( x) + x^T \cdot Q_m \cdot x + a_m \cdot x - b_m^{\mathrm{U}} \Biggr) \cdot \bigl( f_{s}(x) - f_{s}(x)^{\mathrm{UP}} \bigr) \geq0 \\ \displaystyle-1 \cdot \Biggl( \sum _{s = 1}^{S_m} c_{m, s} \cdot f_{m, s}( x) + x^T \cdot Q_m \cdot x + a_m \cdot x - b_m^{\mathrm{U}} \Biggr) \cdot \bigl( f_{s}(x) - f_{s}(x)^{\mathrm{LO}} \bigr) \geq0 \\ \displaystyle-1 \cdot \Biggl( \sum _{s = 1}^{S_m} c_{m, s} \cdot f_{m, s}( x) + x^T \cdot Q_m \cdot x + a_m \cdot x - b_m^{\mathrm{L}} \Biggr) \cdot \bigl( f_{s}(x) - f_{s}(x)^{\mathrm{UP}} \bigr) \geq0 \\ \displaystyle\phantom{-1 \cdot{}} \Biggl( \sum_{s = 1}^{S_m} c_{m, s} \cdot f_{m, s}(x) + x^T \cdot Q_m \cdot x + a_m \cdot x - b_m^{\mathrm{L}} \Biggr) \cdot \bigl( f_{s}(x) - f_{s}(x)^{\mathrm{LO}} \bigr) \geq0 \\ \quad \forall m \in\{ 1, \ldots, M \}; s \in\{ 1, \ldots, S \} \end{array} $$

Nonlinear Term/Nonlinear Term products result in up to four RLT equations:

$$ \begin{array} {ll} \phantom{-1 \cdot{}} \bigl( f_{s}(x) - f_{s}(x)^{\mathrm {UP}} \bigr) \cdot \bigl( f_{s'}(x) - f_{s'}(x)^{\mathrm {UP}} \bigr) \geq0 & \quad\forall s, s' \in\{ 1, \ldots, S \} \\ -1 \cdot \bigl( f_{s}(x) - f_{s}( x)^{\mathrm{UP}} \bigr) \cdot \bigl( f_{s'}(x) - f_{s'}(x)^{\mathrm{LO}} \bigr) \geq0 & \quad\forall s, s' \in\{ 1, \ldots, S \} \\ -1 \cdot \bigl( f_{s}(x) - f_{s}( x)^{\mathrm{LO}} \bigr) \cdot \bigl( f_{s'}(x) - f_{s'}(x)^{\mathrm{UP}} \bigr) \geq0 & \quad\forall s, s' \in\{ 1, \ldots, S \} \\ \phantom{-1 \cdot{}} \bigl( f_{s}(x) - f_{s}(x)^{\mathrm {LO}} \bigr) \cdot \bigl( f_{s'}( x) - f_{s'}(x)^{\mathrm {LO}} \bigr) \geq0 & \quad\forall s, s' \in\{ 1, \ldots, S \} \end{array} $$

Equation/Equation products may add up to four RLT equations:

$$ \begin{aligned} &\Biggl( \sum _{s = 1}^{S_m} c_{m, s} \cdot f_{m, s}( x) + x^T \cdot Q_m \cdot x + a_m \cdot x - b_m^{\mathrm{U}} \Biggr) \\ &\quad\times \Biggl( \sum_{s = 1}^{S_{m'}} c_{s_{m'}} \cdot f_{s_{m'}}(x) + x^T \cdot Q_{m'} \cdot x + a_{m'} \cdot x - b_{m'}^{\mathrm{UP}} \Biggr) \geq0 \\ &-1 \cdot \Biggl( \sum_{s = 1}^{S_m} c_{m, s} \cdot f_{m, s}(x) + x^T \cdot Q_m \cdot x + a_m \cdot x - b_m^{\mathrm{U}} \Biggr) \\ &\quad \times\Biggl( \sum _{s = 1}^{S_{m'}} c_{s_{m'}} \cdot f_{s_{m'}}( x) + x^T \cdot Q_{m'} \cdot x + a_{m'} \cdot x - b_{m'}^{\mathrm{LO}} \Biggr) \geq0 \\ &-1 \cdot \Biggl( \sum_{s = 1}^{S_m} c_{m, s} \cdot f_{m, s}(x) + x^T \cdot Q_m \cdot x + a_m \cdot x - b_m^{\mathrm{L}} \Biggr) \\ &\quad\times \Biggl( \sum _{s = 1}^{S_{m'}} c_{s_{m'}} \cdot f_{s_{m'}}( x) + x^T \cdot Q_{m'} \cdot x + a_{m'} \cdot x - b_{m'}^{\mathrm{UP}} \Biggr) \geq0 \\ & \Biggl( \sum _{s = 1}^{S_m} c_{m, s} \cdot f_{m, s}( x) + x^T \cdot Q_m \cdot x + a_m \cdot x - b_m^{\mathrm{L}} \Biggr) \\ &\quad\times \Biggl( \sum_{s = 1}^{S_{m'}} c_{s_{m'}} \cdot f_{s_{m'}}(x) + x^T \cdot Q_{m'} \cdot x + a_{m'} \cdot x - b_{m'}^{\mathrm{LO}} \Biggr) \geq0 \\ &\quad\forall m, m' \in\{ 1, \ldots, M \} \end{aligned} $$

Appendix B: Recognizing Convexity and Concavity

The MISO framework detects convexity in product \(x_{1}^{a} \cdot x_{2}^{b}\) using the conditions of Maranas and Floudas [36]. Repeating their results without proof:

Theorem Appendix B.1

[36]

If one of the following conditions holds for the product \(x_{1}^{a} \cdot x_{2}^{b}\):

  1. 1.

    x 1,x 2≥0

  2. 2.

    \(a, b \in\mathbb{Z}; a \bmod{2} = 0; b \bmod{2} = 0\)

  3. 3.

    \(a, b \in\mathbb{Z}; a \bmod{2} = 1; b \bmod{2} = 1; x_{1} \cdot x_{2} \geq0\)

  4. 4.

    \(a, b \in\mathbb{Z}; a \bmod{2} = 1; b \bmod{2} = 0; x_{1} \geq0\) (symmetrically amod2=0;bmod2=1;x 2≥0)

then \(x_{1}^{a} \cdot x_{2}^{b}\) is convex if at least two of the following are true: a≤0;b≤0;1−a−b≤0. The term is concave if a,b≥0 and a+b≤1.

Theorem Appendix B.2

[36]

If one of the following conditions holds for the product \(x_{1}^{a} \cdot x_{2}^{b}\):

  1. 1.

    \(a, b \in\mathbb{Z}; a \bmod{2} = 1; b \bmod{2} = 1; x_{1} \cdot x_{2} \leq0\)

  2. 2.

    \(a, b \in\mathbb{Z}; a \bmod{2} = 1; b \bmod{2} = 0; x_{1} \leq0\) (symmetrically amod2=0;bmod2=1;x 2≤0)

then \(x_{1}^{a} \cdot x_{2}^{b}\) is concave if at least two of the following are true: a≤0;b≤0;1−a−b≤0.

For general signomial terms, the MISO framework recognizes convexity/concavity on positive domains:

Theorem Appendix B.3

[36]

Signomial term \(\prod_{c = 1}^{C} x_{c}^{p_{s, c}} : x \in\mathbb {R}^{C}_{+}\) is convex if:

  1. 1.

    p s,c ≤0∀c={1,…,C} or

  2. 2.

    ∃c′ such that p s,c′≥1−∑ c≠c′ p s,c and p s,c ≤0∀c≠c′;c={1,…,C}

and concave if p s,c ≥0∀c={1,…,C};∑ c p s,c ≤1.

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Misener, R., Floudas, C.A. A Framework for Globally Optimizing Mixed-Integer Signomial Programs. J Optim Theory Appl 161, 905–932 (2014). https://doi.org/10.1007/s10957-013-0396-3

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