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Register a new userA Computational Status Update for Exact Rational Mixed Integer Programming
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The last milestone achievement for the roundoff-error-free solution of general mixed integer programs over the rational numbers was a hybrid-precision branch-and-bound algorithm published by Cook, Koch, Steffy, and Wolter in 2013. We describe a substantial revision and extension of this framework that integrates symbolic presolving, features an exact repair step for solutions from primal heuristics, employs a faster rational LP solver based on LP iterative refinement, and is able to produce independently verifiable certificates of optimality. We study the significantly improved performance and give insights into the computational behavior of the new algorithmic components. On the MIPLIB 2017 benchmark set, we observe an average speedup of 6.6x over the original framework and 2.8 times as many instances solved within a time limit of two hours.
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We study robust convex quadratic programs where the uncertain problem parameters can contain both continuous and integer components. Under the natural boundedness assumption on the uncertainty set, we show that the generic problems are amenable to exact copositive programming reformulations of polynomial size. These convex optimization problems are NP-hard but admit a conservative semidefinite programming (SDP) approximation that can be solved efficiently. We prove that the popular approximate S-lemma method --- which is valid only in the case of continuous uncertainty --- is weaker than our approximation. We also show that all results can be extended to the two-stage robust quadratic optimization setting if the problem has complete recourse. We assess the effectiveness of our proposed SDP reformulations and demonstrate their superiority over the state-of-the-art solution schemes on instances of least squares, project management, and multi-item newsvendor problems.
Cutting plane methods play a significant role in modern solvers for tackling mixed-integer programming (MIP) problems. Proper selection of cuts would remove infeasible solutions in the early stage, thus largely reducing the computational burden without hurting the solution accuracy. However, the major cut selection approaches heavily rely on heuristics, which strongly depend on the specific problem at hand and thus limit their generalization capability. In this paper, we propose a data-driven and generalizable cut selection approach, named Cut Ranking, in the settings of multiple instance learning. To measure the quality of the candidate cuts, a scoring function, which takes the instance-specific cut features as inputs, is trained and applied in cut ranking and selection. In order to evaluate our method, we conduct extensive experiments on both synthetic datasets and real-world datasets. Compared with commonly used heuristics for cut selection, the learning-based policy has shown to be more effective, and is capable of generalizing over multiple problems with different properties. Cut Ranking has been deployed in an industrial solver for large-scale MIPs. In the online A/B testing of the product planning problems with more than $10^7$ variables and constraints daily, Cut Ranking has achieved the average speedup ratio of 12.42% over the production solver without any accuracy loss of solution.
We propose a dual dynamic integer programming (DDIP) framework for solving multi-scale mixed-integer model predictive control (MPC) problems. Such problems arise in applications that involve long horizons and/or fine temporal discretizations as well as mixed-integer states and controls (e.g., scheduling logic and discrete actuators). The approach uses a nested cutting-plane scheme that performs forward and backward sweeps along the time horizon to adaptively approximate cost-to-go functions. The DDIP scheme proposed can handle general MPC formulations with mixed-integer controls and states and can perform forward-backward sweeps over block time partitions. We demonstrate the performance of the proposed scheme by solving mixed-integer MPC problems that arise in the scheduling of central heating, ventilation, and air-conditioning (HVAC) plants. We show that the proposed scheme is scalable and dramatically outperforms state-of-the-art mixed-integer solvers.
The most important ingredient for solving mixed-integer nonlinear programs (MINLPs) to global epsilon-optimality with spatial branch and bound is a tight, computationally tractable relaxation. Due to both theoretical and practical considerations, relaxations of MINLPs are usually required to be convex. Nonetheless, current optimization solver can often successfully handle a moderate presence of nonconvexities, which opens the door for the use of potentially tighter nonconvex relaxations. In this work, we exploit this fact and make use of a nonconvex relaxation obtained via aggregation of constraints: a surrogate relaxation. These relaxations were actively studied for linear integer programs in the 70s and 80s, but they have been scarcely considered since. We revisit these relaxations in an MINLP setting and show the computational benefits and challenges they can have. Additionally, we study a generalization of such relaxation that allows for multiple aggregations simultaneously and present the first algorithm that is capable of computing the best set of aggregations. We propose a multitude of computational enhancements for improving its practical performance and evaluate the algorithms ability to generate strong dual bounds through extensive computational experiments.
Small-scale Mixed-Integer Quadratic Programming (MIQP) problems often arise in embedded control and estimation applications. Driven by the need for algorithmic simplicity to target computing platforms with limited memory and computing resources, this paper proposes a few approaches to solving MIQPs, either to optimality or suboptimally. We specialize an existing Accelerated Dual Gradient Projection (GPAD) algorithm to effectively solve the Quadratic Programming (QP) relaxation that arise during Branch and Bound (B&B) and propose a generic framework to warm-start the binary variables which reduces the number of QP relaxations. Moreover, in order to find an integer feasible combination of the binary variables upfront, two heuristic approaches are presented: ($i$) without using B&B, and ($ii$) using B&B with a significantly reduced number of QP relaxations. Both heuristic approaches return an integer feasible solution that may be suboptimal but involve a much reduced computation effort. Such a feasible solution can be either implemented directly or used to set an initial upper bound on the optimal cost in B&B. Through different hybrid control and estimation examples involving binary decision variables, we show that the performance of the proposed methods, although very simple to code, is comparable to that of state-of-the-art MIQP solvers.
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