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.
A Pricing-for-Integrality Primal Heuristic for Branch-and-Price: An Application to the Electric Vehicle Routing Problem with Time Windows and Charging Time Slots
We present a scalable solution method based on an alternating direction method of multipliers and graphics processing units (GPUs) for rapidly computing and tracking a solution of alternating current optimal power flow (ACOPF) problem. Such a fast computation is particularly useful for mitigating the negative impact of frequent load and generation fluctuations on the optimal operation of a large electrical grid. To this end, we decompose a given ACOPF problem by grid components, resulting in a large number of small independent nonlinear nonconvex optimization subproblems. The computation time …
A set of solver-free heuristics for the offshore wind collection system problem are presented. Currently, methods of this type are not able to cope with typical constraints, and most of their variations minimize only for accumulated cable length. The first algorithm is a two-steps decision process, where the output is the design of a tree network satisfying cable thermal limits constraints, but vulnerable to violate planarity constraints. Subsequently, a cable crossings repair heuristic is introduced in order to fix infeasible points from the first heuristic. Finally, a refining heuristic (neg…
Restart techniques are common in gradient-free optimization to deal with multimodal functions. Partial warm restarts are also gaining popularity in gradient-based optimization to improve the rate of convergence in accelerated gradient schemes to deal with ill-conditioned functions. In this paper, we propose a simple warm restart technique for stochastic gradient descent to improve its anytime performance when training deep neural networks. We empirically study its performance on the CIFAR-10 and CIFAR-100 datasets, where we demonstrate new state-of-the-art results at 3.14% and 16.21%, respecti…
It is classical that, when the small deformation is assumed, the incremental analysis problem of an elastoplastic structure with a piecewise-linear yield condition and a linear strain hardening model can be formulated as a convex quadratic programming problem. Alternatively, this paper presents a different formulation, an unconstrained nonsmooth convex optimization problem, and proposes to solve it with an accelerated gradient-like method. Specifically, we adopt an accelerated proximal gradient method, that has been developed for a regularized least squares problem. Numerical experiments show …
Learning to Solve the Quadratic Assignment Problem with Warm-Started MCMC Finetuning
School of Mathematical Science, Peking University（北京大学数学科学学院）；State Key Laboratory of Mathematical Sciences, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院数学科学国家重点实验室）；Beijing International Center for Mathematical Research, Peking University（北京大学北京国际数学研究中心）
In this paper, based on a $Q$-linear convergence analysis and an estimate of the linear convergence factor of the proximal point (PP) algorithm for solving box constrained quadratic programming (BQP) problems, an accelerated proximal point (APP) algorithm for solving BQP problems is presented. To solve the strictly convex BQP problems in each step of the APP algorithm, an efficient homotopy method, which tracks the solution path of a parametric quadratic program, is given. The algorithm with APP algorithm as outer iteration and the homotopy method as inner iteration is named by APP-Hom. The in…
