Multivariate McCormick relaxations

McCormick (Math Prog 10(1):147–175, 1976) provides the framework for convex/concave relaxations of factorable functions, via rules for the product of functions and compositions of the form F ∘ f, where F is a univariate function. Herein, the composition theorem is generalized to allow multivariate outer functions F, and theory for the propagation of subgradients is presented. The generalization interprets the McCormick relaxation approach as a decomposition method for the auxiliary variable method. In addition to extending the framework, the new result provides a tool for the proof of relaxations of specific functions. Moreover, a direct consequence is an improved relaxation for the product of two functions, at least as tight as McCormick’s result, and often tighter. The result also allows the direct relaxation of multilinear products of functions. Furthermore, the composition result is applied to obtain improved convex underestimators for the minimum/maximum and the division of two functions for which current relaxations are often weak. These cases can be extended to allow composition of a variety of functions for which relaxations have been proposed

Chebyshev model arithmetic for factorable functions

This article presents an arithmetic for the computation of Chebyshev models for factorable functions and an analysis of their convergence properties. Similar to Taylor models, Chebyshev models consist of a pair of a multivariate polynomial approximating the factorable function and an interval remainder term bounding the actual gap with this polynomial approximant. Propagation rules and local convergence bounds are established for the addition, multiplication and composition operations with Chebyshev models. The global convergence of this arithmetic as the polynomial expansion order increases is also discussed. A generic implementation of Chebyshev model arithmetic is available in the library MC++. It is shown through several numerical case studies that Chebyshev models provide tighter bounds than their Taylor model counterparts, but this comes at the price of extra computational burden

Global Optimization of Gaussian processes

Gaussian processes~(Kriging) are interpolating data-driven models that are frequently applied in various disciplines. Often, Gaussian processes are trained on datasets and are subsequently embedded as surrogate models in optimization problems. These optimization problems are nonconvex and global optimization is desired. However, previous literature observed computational burdens limiting deterministic global optimization to Gaussian processes trained on few data points. We propose a reduced-space formulation for deterministic global optimization with trained Gaussian processes embedded. For optimization, the branch-and-bound solver branches only on the degrees of freedom and McCormick relaxations are propagated through explicit Gaussian process models. The approach also leads to significantly smaller and computationally cheaper subproblems for lower and upper bounding. To further accelerate convergence, we derive envelopes of common covariance functions for GPs and tight relaxations of acquisition functions used in Bayesian optimization including expected improvement, probability of improvement, and lower confidence bound. In total, we reduce computational time by orders of magnitude compared to state-of-the-art methods, thus overcoming previous computational burdens. We demonstrate the performance and scaling of the proposed method and apply it to Bayesian optimization with global optimization of the acquisition function and chance-constrained programming. The Gaussian process models, acquisition functions, and training scripts are available open-source within the "MeLOn - Machine Learning Models for Optimization" toolbox~(https://git.rwth-aachen.de/avt.svt/public/MeLOn)

Convex Hull Formulations for Mixed-Integer Multilinear Functions

In this paper, we present convex hull formulations for a mixed-integer, multilinear term/function (MIMF) that features products of multiple continuous and binary variables. We develop two equivalent convex relaxations of an MIMF and study their polyhedral properties in their corresponding higher-dimensional spaces. We numerically observe that the proposed formulations consistently perform better than state-of-the-art relaxation approaches

Linear Programming Relaxations of Quadratically Constrained Quadratic Programs

We investigate the use of linear programming tools for solving semidefinite programming relaxations of quadratically constrained quadratic problems. Classes of valid linear inequalities are presented, including sparse PSD cuts, and principal minors PSD cuts. Computational results based on instances from the literature are presented.Comment: Published in IMA Volumes in Mathematics and its Applications, 2012, Volume 15

Global Deterministic Optimization with Artificial Neural Networks Embedded

Artificial neural networks (ANNs) are used in various applications for data-driven black-box modeling and subsequent optimization. Herein, we present an efficient method for deterministic global optimization of ANN embedded optimization problems. The proposed method is based on relaxations of algorithms using McCormick relaxations in a reduced-space [\textit{SIOPT}, 20 (2009), pp. 573-601] including the convex and concave envelopes of the nonlinear activation function of ANNs. The optimization problem is solved using our in-house global deterministic solver MAiNGO. The performance of the proposed method is shown in four optimization examples: an illustrative function, a fermentation process, a compressor plant and a chemical process optimization. The results show that computational solution time is favorable compared to the global general-purpose optimization solver BARON.Comment: J Optim Theory Appl (2018