The 10 most recently published documents
Doubly nonnegative (DNN) relaxation usually provides a tight lower bound for a
mixed-binary quadratic program (MBQP). However, solving DNN problems is chal-
lenging because: (1) the problem size is ((n + l)2) for an MBQP with n variables
and l inequality constraints, and (2) the rank of optimal solutions cannot be estimated
a priori due to the absence of theoretical bounds. In this work, we propose RiNNAL+,
a Riemannian augmented Lagrangian method (ALM) for solving DNN problems. We
prove that the DNN relaxation of an MBQP, with matrix dimension (n + l + 1), is
equivalent to the SDP-RLT relaxation (based on the reformulation-linearization tech-
nique) with a smaller matrix dimension (n + 1). In addition, we develop a hybrid
method that alternates between two phases to solve the ALM subproblems. In phase
one, we apply low-rank matrix factorization and random perturbation to transform the
feasible region into a lower-dimensional manifold so that we can use the Riemannian
gradient descent method. In phase two, we apply a single projected gradient step to
update the rank of the underlying variable and escape from spurious local minima
arising in the first phase if necessary. To reduce the computation cost of the projected
gradient step, we develop pre-processing and warm-start techniques for acceleration.
Unlike traditional rank-adaptive methods that require extensive parameter tuning, our
hybrid method requires minimal tuning. Extensive experiments confirm the efficiency
and robustness of RiNNAL+ in solving various classes of large-scale DNN problems
In this paper, we introduce a novel semidefinite programming framework for designing custom frugal resolvent splitting algorithms which find a zero in the sum of n monotone operators. This framework features a number of design choices which facilitate creating resolvent splitting algorithms with specific communication structure. We illustrate these design choices using a variety of constraint sets and objective functions, as well as the use of a mixed-integer SDP to minimize time per iteration or required number of communications between the resolvents which define the splitting. Using the Performance Estimation Problem (PEP) framework, we provide parameter selections, such as step size, which for high dimensional problems provide optimal contraction factors in the algorithms. Among the algorithm design choices we introduce, we provide a characterization of algorithm designs which provide minimal convergence time given structural properties of the monotone operators, resolvent computation times, and communication latencies.
Adjustable robust optimization (ARO) is a powerful tool to model problems that have uncertain data and that feature a two-stage decision-making process. Computationally, they are often addressed using the column-and-constraint generation (CCG) algorithm introduced by Zeng and Zhao [45]. While it was empirically shown that the algorithm scales well if all second-stage decisions are continuous, the presence of integer variables in the second stage rapidly leads to challenging large-scale mixed-integer problems within CCG. These problems can no longer be solved to global optimality within reasonable time limits in general. In this work, we explicitly focus on ARO problems with mixed-integer second-stage decisions and discuss the main difficulties of successfully applying CCG to this problem class. We then introduce, for a large set of problems with specific structural properties, a stronger formulation, which can be used in place of the master problem in the classic CCG algorithm. We show how this model can be effectively solved by column generation (CG). Additionally, we introduce a new CG-based heuristic that is able to generate new feasible points to speed up the overall method. We apply this nested scheme, combining CCG and CG, to three problems from logistics and scheduling to show the applicability of our approach.
Support matrix machine (SMM) is a successful supervised classification model for matrix-type samples. Unlike support vector machines, it employs low-rank regularization on the regression matrix to effectively capture the intrinsic structure embedded in each input matrix. When solving a large-scale SMM, a major challenge arises from the potential increase in sample size, leading to substantial computational and storage burdens. To address these issues, we design a semismooth Newton-CG (SNCG) based augmented Lagrangian method (ALM) for solving the SMM. The ALM exhibits an asymptotic R-superlinear convergence if a strict complementarity condition is satisfied. The SNCG method is employed to solve the ALM subproblems, achieving at least a superlinear convergence rate under the nonemptiness of an index set. Furthermore, the sparsity of samples and the low-rank nature of solutions enable us to reduce the computational cost and storage demands for the Newton linear systems. Additionally, we develop an adaptive sieving strategy that generates a solution path for the SMM by exploiting sample sparsity. The finite convergence of this strategy is also demonstrated. Numerical experiments on both large-scale real and synthetic datasets validate the effectiveness of the proposed methods.
A conflict graph represents logical relations between binary variables, and effective use of the graph can significantly accelerate branch-and-cut solvers for mixed-integer programming (MIP). In this paper we develop efficient parallel conflict graph management: conflict detection; maximal clique generation; clique extension; and clique merging. We leverage parallel computing in order to intensify computational effort on the conflict graph, thereby generating a much larger pool of cutting planes than what can be practically achieved in serial. Computational experiments demonstrate that the expanded pool of cuts enabled by parallel computing lead to substantial reductions in total MIP solve time, especially for more challenging cases.
The Standard Quadratic optimization Problem (StQP), arguably the simplest among all classes of NP-hard optimization problems, consists of extremizing a quadratic form (the simplest nonlinear polynomial) over the standard simplex (the simplest polytope/compact feasible set). As a problem class, StQPs may be nonconvex with an exponential number of inefficient local solutions. StQPs arise in a multitude of applications, among them mathematical finance, machine learning (clustering), and modeling in biosciences (e.g., selection and ecology). This paper deals with such StQPs under an additional sparsity or cardinality constraint, which, even for convex objectives, renders NP-hard problems. One motivation to study StQPs under such sparsity restrictions is the high-dimensional portfolio selection problem with too many assets to handle, in particular, in the presence of transaction costs. Here, relying on modern conic optimization techniques, we present tractable convex relaxations for this relevant but difficult problem. We propose novel equivalent reformulations of these relaxations with significant dimensional reduction, which is essential for the tractability of these relaxations when the problem size grows. Moreover, we propose an instance generation procedure which systematically avoids too easy instances. Our extensive computational results illustrate the high quality of the relaxation bounds in a significant number of instances. Furthermore, in contrast with exact mixed-integer quadratic programming models, the solution time of the relaxations is very robust to the choices of the problem parameters. In particular, the reduced formulations achieve significant improvements in terms of the solution time over their counterparts.
We optimize the running time of the primal-dual algorithms by optimizing their stopping criteria for solving convex optimization problems under affine equality constraints, which means terminating the algorithm earlier with fewer iterations. We study the relations between four stopping criteria and show under which conditions they are accurate to detect optimal solutions. The uncomputable one: "Optimality gap and Feasibility error", and the computable ones: the "Karush–Kuhn–Tucker error", the "Projected Duality Gap", and the "Smoothed Duality Gap". Assuming metric sub-regularity or quadratic error bound, we establish that all of the computable criteria provide practical upper bounds for the optimality gap, and approximate it effectively. Furthermore, we establish comparability between some of the computable criteria under certain conditions. Numerical experiments on basis pursuit, and quadratic programs with(out) non-negative weights corroborate these findings and show that the smoothed duality gap is more widely applicable than the rest.
A caterpillar is a connected graph such that the removal of all its vertices with degree results in a path. Given a graph G, a caterpillar-packing of G is a set of vertex-disjoint (not necessarily induced) subgraphs of G such that each subgraph is a caterpillar. In this work we consider the set of caterpillar-packings of a graph, which corresponds to feasible solutions of the 2-schemes strip cutting problem with a sequencing constraint (2-SSCPsc) presented by Rinaldi and Franz (Eur J Oper Res 183:1371–1384, 2007) We show new facet-preserving procedures for this polytope, and we present a general cut-generating strategy based on these procedures. Computational experiments show that this approach is effective in practice.
Mixed-integer nonlinear optimization encompasses a broad class of problems that
present both theoretical and computational challenges. We propose a new type of
method to solve these problems based on a branch-and-bound algorithm with convex node relaxations. These relaxations are solved with a Frank–Wolfe algorithm over the convex hull of mixed-integer feasible points instead of the continuous relaxation via calls to a mixed-integer linear solver as the linear minimization oracle. The proposed method computes feasible solutions while working on a single representation of the polyhedral constraints, leveraging the full extent of mixed-integer linear solvers without an outer approximation scheme and can exploit inexact solutions of node subproblems.
We propose a reference framework to apply the multiparent path relinking (MPR) methodology. MPR is an extension of path relinking (PR) that has been proposed and described but, to the best of our knowledge, it has never been applied. PR is a trajectory-based neighborhood search strategy that explores paths that traverse pairs of solutions. PR has been widely used for search intensification purposes within metaheuristic implementations. To show how MPR can be embedded in more than one setting, our proposal consists of both employing MPR as a post-processing step in a greedy randomized adaptive search procedure (GRASP) and as the combination method in a scatter search (SS). We use the power dominating set problem (PDSP) as our testing platform because its difficulty and structure enable the design and assessment of various strategies. The PDSP seeks to find the minimum placement of measurement devices in an electrical network to monitor the entire system. Our computational experiments are designed to identify the contribution of each element of our proposed MRP implementations.