Search

Recent advances in the efficiency and robustness of algorithms solving convex quadratically constrained quadratic programming (QCQP) problems motivate developing techniques for creating convex quadratic relaxations that, although more expensive to compute, provide tighter bounds than their classical linear counterparts. In the first part of this two-paper series [Strahl et al., 2024], we developed a cutting plane algorithm to construct convex quadratic underestimators for twice-differentiable convex functions, which we extend here to address the case of non-convex difference-of-convex (d.c.) functions as well. Furthermore, we generalize our approach to consider a hierarchy of quadratic forms, thereby allowing the construction of even tighter underestimators. On a set of d.c. functions extracted from benchmark libraries, we demonstrate noteworthy reduction in the hypervolume between our quadratic underestimators and linear ones constructed at the same points. Additionally, we construct convex QCQP relaxations at the root node of a spatial branch-and-bound tree for a set of systematically created d.c. optimization problems in up to four dimensions, and we show that our relaxations reduce the gap between the lower bound computed by the state-of-the-art global optimization solver BARON and the optimal solution by an excess of 90%, on average.

When computing bounds, spatial branch-and-bound algorithms often linearly outer approximate convex relaxations for non-convex expressions in order to capitalize on the efficiency and robustness of linear programming solvers. Considering that linear outer approximations sacrifice accuracy when approximating highly nonlinear functions and recognizing the recent advancements in the efficiency and robustness of available methods to solve optimization problems with quadratic objectives and constraints, we contemplate here the construction of quadratic outer approximations of twice-differentiable convex functions for use in deterministic global optimization. To this end, we present a novel cutting-plane algorithm that determines the tightest scaling parameter, $\alpha$, in the second-order Taylor series approximation quadratic underestimator proposed by Su et al. We use a representative set of convex functions extracted from optimization benchmark libraries to showcase--qualitatively and quantitatively--the tightness of the constructed quadratic underestimators and to demonstrate the overall computational efficiency of our algorithm. Furthermore, we extend our construction procedure to generate even tighter quadratic underestimators by allowing overestimation in infeasible polyhedral regions of optimization problems, as informed by the latter's linear constraints.

Gas networks are used to transport natural gas, which is an important resource for both residential and industrial customers throughout the world. The gas network design problem is generally modelled as a nonconvex mixed-integer nonlinear integer programming problem (MINLP). The challenges of solving the resulting MINLP arise due to the nonlinearity and nonconvexity. In this paper, we propose a framework to study the "design variant" of the problem in which the variables are the diameter choices of the pipes, the flows, the potentials, and the states of various network components. We utilize a nested loop that includes a two-stage procedure that involves a convex reformulation of the original problem in the inner loop and an efficient enumeration scheme in the outer loop. We conduct experiments on benchmark networks to validate and analyze the performance of our framework.

This paper investigates the effectiveness of transfer learning based on Mallows' Cp. We propose a procedure that combines transfer learning with Mallows' Cp (TLCp) and prove that it outperforms the conventional Mallows' Cp criterion in terms of accuracy and stability. Our theoretical results indicate that, for any sample size in the target domain, the proposed TLCp estimator performs better than the Cp estimator by the mean squared error (MSE) metric in the case of orthogonal predictors, provided that i) the dissimilarity between the tasks from source domain and target domain is small, and ii) the procedure parameters (complexity penalties) are tuned according to certain explicit rules. Moreover, we show that our transfer learning framework can be extended to other feature selection criteria, such as the Bayesian information criterion. By analyzing the solution of the orthogonalized Cp, we identify an estimator that asymptotically approximates the solution of the Cp criterion in the case of non-orthogonal predictors. Similar results are obtained for the non-orthogonal TLCp. Finally, simulation studies and applications with real data demonstrate the usefulness of the TLCp scheme., Comment: Accepted to the Journal of Machine Learning Research

We consider the global optimization of nonconvex mixed-integer quadratic programs with linear equality constraints. In particular, we present a new class of convex quadratic relaxations which are derived via quadratic cuts. To construct these quadratic cuts, we solve a separation problem involving a linear matrix inequality with a special structure that allows the use of specialized solution algorithms. Our quadratic cuts are nonconvex, but define a convex feasible set when intersected with the equality constraints. We show that our relaxations are an outer-approximation of a semi-infinite convex program which under certain conditions is equivalent to a well-known semidefinite program relaxation. The new relaxations are implemented in the global optimization solver BARON, and tested by conducting numerical experiments on a large collection of problems. Results demonstrate that, for our test problems, these relaxations lead to a significant improvement in the performance of BARON., Comment: 28 pages

Sparse linear regression is a vast field and there are many different algorithms available to build models. Two new papers published in Statistical Science study the comparative performance of several sparse regression methodologies, including the lasso and subset selection. Comprehensive empirical analyses allow the researchers to demonstrate the relative merits of each estimator and provide guidance to practitioners. In this discussion, we summarize and compare the two studies and we examine points of agreement and divergence, aiming to provide clarity and value to users. The authors have started a highly constructive dialogue, our goal is to continue it., Comment: 12 pages

We consider the global optimization of nonconvex quadratic programs and mixed-integer quadratic programs. We present a family of convex quadratic relaxations which are derived by convexifying nonconvex quadratic functions through perturbations of the quadratic matrix. We investigate the theoretical properties of these quadratic relaxations and show that they are equivalent to some particular semidefinite programs. We also introduce novel branching variable selection strategies which can be used in conjunction with the quadratic relaxations investigated in this paper. We integrate the proposed relaxation and branching techniques into the global optimization solver BARON, and test our implementation by conducting numerical experiments on a large collection of problems. Results demonstrate that the proposed implementation leads to very significant reductions in BARON's computational times to solve the test problems., Comment: 33 pages, 26 figures

Reinforcement learning (RL) has been widely applied to game-playing and surpassed the best human-level performance in many domains, yet there are few use-cases in industrial or commercial settings. We introduce OR-Gym, an open-source library for developing reinforcement learning algorithms to address operations research problems. In this paper, we apply reinforcement learning to the knapsack, multi-dimensional bin packing, multi-echelon supply chain, and multi-period asset allocation model problems, as well as benchmark the RL solutions against MILP and heuristic models. These problems are used in logistics, finance, engineering, and are common in many business operation settings. We develop environments based on prototypical models in the literature and implement various optimization and heuristic models in order to benchmark the RL results. By re-framing a series of classic optimization problems as RL tasks, we seek to provide a new tool for the operations research community, while also opening those in the RL community to many of the problems and challenges in the OR field., Comment: 29 pages, 10 figures

Optimization solvers routinely utilize presolve techniques, including model simplification, reformulation and domain reduction techniques. Domain reduction techniques are especially important in speeding up convergence to the global optimum for challenging nonconvex nonlinear programming (NLP) and mixed-integer nonlinear programming (MINLP) optimization problems. In this work, we survey the various techniques used for domain reduction of NLP and MINLP optimization problems. We also present a computational analysis of the impact of these techniques on the performance of various widely available global solvers on a collection of 1740 test problems.

Simulation Optimization (SO) refers to the optimization of an objective function subject to constraints, both of which can be evaluated through a stochastic simulation. To address specific features of a particular simulation---discrete or continuous decisions, expensive or cheap simulations, single or multiple outputs, homogeneous or heterogeneous noise---various algorithms have been proposed in the literature. As one can imagine, there exist several competing algorithms for each of these classes of problems. This document emphasizes the difficulties in simulation optimization as compared to mathematical programming, makes reference to state-of-the-art algorithms in the field, examines and contrasts the different approaches used, reviews some of the diverse applications that have been tackled by these methods, and speculates on future directions in the field.

ALAMO is a computational methodology for leaning algebraic functions from data. Given a data set, the approach begins by building a low-complexity, linear model composed of explicit non-linear transformations of the independent variables. Linear combinations of these non-linear transformations allow a linear model to better approximate complex behavior observed in real processes. The model is refined, as additional data are obtained in an adaptive fashion through error maximization sampling using derivative-free optimization. Models built using ALAMO can enforce constraints on the response variables to incorporate first-principles knowledge. The ability of ALAMO to generate simple and accurate models for a number of reaction problems is demonstrated. The error maximization sampling is compared with Latin hypercube designs to demonstrate its sampling efficiency. ALAMO's constrained regression methodology is used to further refine concentration models, resulting in models that perform better on validation data and satisfy upper and lower bounds placed on model outputs.

Optimization of the refrigerant circuitry can improve a heat exchanger's performance. Design engineers currently choose the refrigerant circuitry according to their experience and heat exchanger simulations. However, the design of an optimized refrigerant circuitry is difficult. The number of refrigerant circuitry candidates is enormous. Therefore, exhaustive search algorithms cannot be used and intelligent techniques must be developed to explore the solution space efficiently. In this paper, we formulate refrigerant circuitry design as a binary constrained optimization problem. We use CoilDesigner, a simulation and design tool of air to refrigerant heat exchangers, in order to simulate the performance of different refrigerant circuitry designs. We treat CoilDesigner as a black-box system since the exact relationship of the objective function with the decision variables is not explicit. Derivative-free optimization (DFO) algorithms are suitable for solving this black-box model since they do not require explicit functional representations of the objective function and the constraints. The aim of this paper is twofold. First, we compare four mixed-integer constrained DFO solvers and one box-bounded DFO solver and evaluate their ability to solve a difficult industrially relevant problem. Second, we demonstrate that the proposed formulation is suitable for optimizing the circuitry configuration of heat exchangers. We apply the DFO solvers to 17 heat exchanger design problems. Results show that TOMLAB/glcDirect and TOMLAB/glcSolve can find optimal or near-optimal refrigerant circuitry designs after a relatively small number of circuit simulations.

This article provides an introduction to and review of the field of computer-aided molecular design (CAMD). It is intended to be approachable for the absolute beginner as well as useful to the seasoned CAMD practitioner. We begin by discussing various quantitative structure-property relationships (QSPRs) which have been demonstrated to work well with CAMD problems. The methods discussed in this article are (1) group contribution methods, (2) topological indices, and (3) signature descriptors. Next, we present general optimization formulations for various forms of the CAMD problem. Common design constraints are discussed and structural feasibility constraints are provided for the three types of QSPRs addressed. We then detail useful techniques for approaching CAMD optimization problems, including decomposition methods, heuristic approaches, and mathematical programming strategies. Finally, we discuss many applications that have been addressed using CAMD., Comment: 38 pages, 13 figures, 3 tables, 173 references