Linear Programming Relaxation Research Articles

An Integer Programming Approach to Subspace Clustering with Missing Data

In the subspace clustering with missing data (SCMD) problem, we are given a collection of n partially observed d-dimensional vectors. The data points are assumed to be concentrated near a union of low-dimensional subspaces. The goal of SCMD is to cluster the vectors according to their subspace membership and recover the underlying basis, which can then be used to infer their missing entries. State-of-the-art algorithms for SCMD can fail on instances with a high proportion of missing data, with full-rank data, or if the underlying subspaces are similar to each other. We propose a novel integer programming approach for SCMD. The approach is based on dynamically determining a set of candidate subspaces and optimally assigning points to selected subspaces. The problem structure is identical to the classical facility-location problem, with subspaces playing the role of facilities and data points playing that of customers. We propose a column-generation approach for identifying candidate subspaces combined with a Benders decomposition approach for solving the linear programming relaxation of the formulation. An empirical study demonstrates that the proposed approach can achieve better clustering accuracy than state-of-the-art methods when the data are high rank, the percentage of missing data is high, or the subspaces are similar. Funding: Support for this research was provided by American Family Insurance through a research partnership with the University of Wisconsin–Madison’s Data Science Institute. Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoo.2023.0020 .

A fix-propagate-repair heuristic for mixed integer programming

AbstractWe describe a diving heuristic framework based on constraint propagation for mixed integer linear programs. The proposed approach is an extension of the common fix-and-propagate scheme, with the addition of solution repairing after each step. The repair logic is loosely based on the WalkSAT strategy for boolean satisfiability. Different strategies for variable ranking and value selection, as well as other options, yield different diving heuristics. The overall method is relatively inexpensive, as it is basically LP-free: the full linear programming relaxation is solved only at the beginning (and only for the ranking strategies that make use of it), while additional, typically much smaller, LPs are only used to compute values for the continuous variables (if any), once at the bottom of a dive. While individual strategies are not very robust in finding feasible solutions on a heterogeneous testbed, a portfolio approach proved quite effective. In particular, it could consistently find feasible solutions in 189 out of 240 instances from the public MIPLIB 2017 benchmark testbed, in a matter of a few seconds of runtime. The framework has also been implemented inside the commercial MIP solver Xpress and shown to give a small performance improvement in time to optimality on a large internal heterogeneous testbed.

Maritime Inventory Routing with Speed Optimization: A MIQCP Formulation for a Tanker Fleet Servicing FPSO Units

This paper presents a continuous-time formulation for the management of deep-sea floating production storage and offloading (FPSO) units, which process and store crude oil from nearby oil platforms while waiting for a shuttle tanker to arrive and collect it. A heterogeneous fleet of tanker vessels travelling between the FPSO units and a port refinery is available, with the optimization deciding on the number and type of vessels to use, their route and travelling speed. The goal is to achieve an environmentally friendly solution featuring vessels travelling at lower speeds to minimize fuel consumption (a quadratic function of speed), which may require renting additional shuttle tankers to maintain production. The resulting mixed-integer quadratically constrained problems (MIQCP) can be solved by GUROBI, but not to global optimality, whereas a mixed-integer linear programming (MILP) relaxation that underestimates the total operating cost can tackle moderate problem sizes (5 FPSOs and 4 shuttle tankers).

Optimization of IoT perceived content caching in F-RANs: Minimum retrieval delay and resource extension with performance sensitivity

In the Internet of Things (IoT) perceived applications of monitoring the states of the environment, a feasible technology is to use fog radio access networks (F-RANs) to alleviate the problems of long response time and cloud server bottlenecks in cloud computing. In response to the above problems, this work investigates the problem of minimizing the retrieval delay of IoT contents in F-RANs under the constraints of system resources. The problem is formulated as an integer linear programming (ILP) model. Then, a polynomial-time method with linear programming (LP) relaxation and rounding is proposed to approximate the optimal solution of the problem. Through proof, the method can obtain a feasible solution with a bounded approximation ratio in polynomial time. The conducted simulations validate that the obtained feasible solution is very close to the optimal one. On the other hand, when the system resources are not enough to meet the continuous growth of content retrieval and need to be expanded, this work further establishes an association relation between cached contents and system resources. Based on the above relation, the second method of expanding system resources with performance sensitivity is proposed to provide the service provider with an effective and economical expansion of system resources. It utilizes a predefined system parameter in balancing the trade-off between the approximation ratio to the optimal solution of the problem and the extended system resources. The solution obtained by the second method is also proved to have a bounded approximation ratio.

The circle packing problem: A theoretical comparison of various convexification techniques

We consider the problem of packing a given number of congruent circles with the maximum radius in a unit square as a mathematical optimization problem. Due to the presence of non-overlapping constraints, this problem is a notoriously difficult nonconvex quadratically constrained optimization problem, which possesses many local optima. We consider several popular convexification techniques, giving rise to linear programming relaxations and semidefinite programming relaxations for the circle packing problem. We compare the strength of these relaxations theoretically, thereby proving the conjectures by Anstreicher. Our results serve as a theoretical justification for the ineffectiveness of existing machinery for convexification of non-overlapping constraints.

An outcome space algorithm for solving general linear multiplicative programming

The following article presents and corroborates an outcome space branch-and-bound algorithm for solving the general linear multiplicative programming problem (GLMPP). In this new algorithm, GLMPP is transformed into its equivalent problem by introducing auxiliary variables. To compute the tight upper bound for the optimal value of GLMPP, the linear relaxation programming problem is assembled by using bound and hull linear approximations. Furthermore, the global convergence and computational complexity of the algorithm are demonstrated. Finally, the soundness and advantage of the proposed algorithm are validated by solving a demonstrative linear multiplicative problem.

Integrated aircraft routing and cargo routing problem for combination airlines

The combination airlines operate both passenger aircraft and freighter aircraft to meet passenger and cargo demand. At present, combination airlines employ a sequential approach to allocating their capacity for passenger and cargo demand. Nevertheless, implementing an integrated resource allocation procedure has the potential to improve overall resource allocation efficiency. In this paper, we introduce an integrated model to help combination airlines integrate their aircraft routing and cargo routing decisions to maximize the expected overall profits derived from both passenger and cargo demand. We considered the stochastic nature of passenger baggage and proposed a set of individual chance constraints to ensure the robustness of the integrated solution. We reformulate the chance constraints using piecewise linear approximation to ensure solution efficiency. In addition, we proposed a column-and-row generation based solution approach that removes the through-connection related constraints at the beginning of the solution process and then adds the columns and rows during the iterations as needed. We proved that the proposed column-and-row generation approach can obtain an optimal solution for the LP relaxation problem. The model and the solution approach were tested in a number of scenarios obtained from a major Chinese combination airline. The computational results show that the combination airline can improve their expected profits by integrating capacity allocation. The results also demonstrated that the proposed column-and-row generation solution approach can decrease the solution time of the integrated model. These findings indicate that the model and the solution method are useful and efficient tools for combination airlines when planning their aircraft and cargo routes.

A tight formulation for the dial-a-ride problem

Ridepooling services play an increasingly important role in modern transportation systems. With soaring demand and growing fleet sizes, the underlying route planning problems become increasingly challenging. In this context, we consider the dial-a-ride problem (DARP): Given a set of transportation requests with pick-up and delivery locations, passenger numbers, time windows, and maximum ride times, an optimal routing for a fleet of vehicles, including an optimized passenger assignment, needs to be determined. We present tight mixed-integer linear programming (MILP) formulations for the DARP by combining two state-of-the-art models into novel location-augmented-event-based formulations. Strong valid inequalities and lower and upper bounding techniques are derived to further improve the formulations. We then demonstrate the theoretical and computational superiority of the new models: First, the linear programming relaxations of the new formulations are stronger than existing location-based approaches. Second, extensive numerical experiments on benchmark instances show that computational times are on average reduced by 53.9% compared to state-of-the-art event-based approaches.

An optimization framework for solving large scale multidemand multidimensional knapsack problem instances employing a novel core identification heuristic

By applying the core concept to solve a binary integer program (BIP), certain variables of the BIP are fixed to their anticipated values in the optimal solution. In contrast, the remaining variables, called core variables, are used to construct and solve a core problem (CP) instead of the BIP. A new approach for identifying CP utilizing a local branching (LB) alike constraint is presented in this article. By including the LB-like constraint in the linear programming relaxation of the BIP, this method transfers batches of variables to the set of core variables by analyzing changes to their reduced costs. This approach is sensitive to problem hardness because more variables are moved to the core set for hard problems compared to easy ones. This novel core identification approach is embedded in a multi-stage framework to solve the multidemand, multidimensional knapsack problems (MDMKP), where at each stage, more variables are added to the previous stage CP. The default branch and bound of CPLEX20.10 is used to solve the first stage, and a tabu search algorithm is used to solve subsequent stages until all variables are added to CP in the last stage. The new framework has shown equivalent to superior results compared to the state-of-the-art algorithms in solving large MDMKP instances having 500 and 1,000 variables.

Star Bicolouring of Bipartite Graphs

We give an integer linear program formulation for the star bicolouring of bipartite graphs. We develop a column generation method to solve the linear programming relaxation to obtain a lower bound for the minimum number of colours needed. We determine the star bicolouring using the iterative rounding method. We give computational results on arrowhead matrices, sparse random matrices, complete bipartite graphs, and matrices from the Harwell–Boeing collection. The findings demonstrate that the proposed method effectively establishes lower and upper bounds for the minimum number of colours needed for a star bicolouring of bipartite graphs, particularly for sparse bipartite graphs.

Maximizing Engagement in Large-Scale Social Networks

Motivated by the importance of user engagement as a crucial element in cascading leaving of users from a social network, we study identifying a largest relaxed variant of a degree-based cohesive subgraph: the maximum anchored k-core problem. Given graph [Formula: see text] and integers k and b, the maximum anchored k-core problem seeks to find a largest subset of vertices [Formula: see text] that induces a subgraph with at least [Formula: see text] vertices of degree at least k. We introduce a new integer programming (IP) formulation for the maximum anchored k-core problem and conduct a polyhedral study on the polytope of the problem. We show the linear programming relaxation of the proposed IP model is at least as strong as that of a naïve formulation. We also identify facet-defining inequalities of the IP formulation. Furthermore, we develop inequalities and fixing procedures to improve the computational performance of our IP model. We use benchmark instances to compare the computational performance of the IP model with (i) the naïve IP formulation and (ii) two existing heuristic algorithms. Our proposed IP model can optimally solve half of the benchmark instances that cannot be solved to optimality either by the naïve model or the existing heuristic approaches. Funding: This work is funded by the National Science Foundation (NSF) [Grant DMS-2318790] titled AMPS: Novel Combinatorial Optimization Techniques for Smartgrids and Power Networks. Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoo.2022.0024 .

On the Complexity of Symmetric vs. Functional PCSPs

The complexity of the promise constraint satisfaction problem \(\operatorname{(PCSP)}(\mathbf{A},\mathbf{B})\) is largely unknown, even for symmetric \(\mathbf{A}\) and \(\mathbf{B}\) , except for the case when \(\mathbf{A}\) and \(\mathbf{B}\) are Boolean. First, we establish a dichotomy for \(\operatorname{PCSP}(\mathbf{A},\mathbf{B})\) where \(\mathbf{A},\mathbf{B}\) are symmetric, \(\mathbf{B}\) is functional (i.e., any \(r-1\) elements of an \(r\) -ary tuple uniquely determines the last one), and \((\mathbf{A},\mathbf{B})\) satisfies technical conditions we introduce called dependency and additivity . This result implies a dichotomy for \(\operatorname{PCSP}(\mathbf{A},\mathbf{B})\) with \(\mathbf{A},\mathbf{B}\) symmetric and \(\mathbf{B}\) functional if (i) \(\mathbf{A}\) is Boolean, or (ii) \(\mathbf{A}\) is a hypergraph of a small uniformity, or (iii) \(\mathbf{A}\) has a relation \(R^{\mathbf{A}}\) of arity at least three such that the hypergraph diameter of \((A,R^{\mathbf{A}})\) is at most one. Second, we show that for \(\operatorname{PCSP}(\mathbf{A},\mathbf{B})\) , where \(\mathbf{A}\) and \(\mathbf{B}\) contain a single relation, \(\mathbf{A}\) satisfies a technical condition called balancedness , and \(\mathbf{B}\) is arbitrary, the combined basic linear programming relaxation and the affine integer programming (AIP) relaxation is no more powerful than the (in general strictly weaker) \({{\rm AIP}}\) relaxation. Balanced \(\mathbf{A}\) include symmetric \(\mathbf{A}\) or, more generally, \(\mathbf{A}\) preserved by a transitive permutation group.

A multi‐objective sustainable closed‐loop supply chain network problem with hybrid facilities

AbstractA sustainable closed‐loop supply chain network requires conjunctive implementation of reverse logistics in the supply chain, with decisions that consider economic, environmental, and social factors. In real life, the problem needs to be addressed by prioritizing targets or interacting between them to give a range of solutions to the decision maker. In this context, this work proposes a novel multi‐objective sustainable closed‐loop supply chain network problem based on the revised network design model with hybrid recovery centers minimizing (1) the total economic cost, (2) the CO2 emission of vehicles used, and (3) the total obnoxious distance. The latter objective is a novel implementation of the social dimension of a sustainable model. A sensitivity analysis of the multi‐objective model is developed through ANOVA. A dataset of instances was generated to test the model and the solution methods, which are configured with AUGMECON2, a linear programming relaxation implemented to improve the CPU time, and AUGMECON2‐EXTENDED to obtain more solutions to avoid exploring all space of the solution. The results show that an AUGMECON2‐EXTENDED implementation outperforms all the selected performance metrics. These performance metrics include NPS, CPU time, RPOS, QM, and HV. The results show an improvement on average of at least , , , , and , respectively, in those metrics, in comparison to other implementations.

A better LP rounding for feedback arc set on tournaments

We present a randomized algorithm to approximate the feedback arc set problem on weighted tournaments, a classic well-studied NP-hard problem. Our algorithm is based on rounding its standard linear programming relaxation. It improves the previously best-known LP rounding algorithm by achieving an approximation factor of 2.127 (best known was 2.5). As a result, we have found a better upper bound for the integrality gap of the corresponding LP.

Polynomial Integrality Gap of Flow LP for Directed Steiner Tree

In the Directed Steiner Tree (DST) problem, we are given a directed graph \(G=(V,E)\) on \(n\) vertices with edge-costs \(c\in{\mathbb{R}}_{\geq 0}^{E}\) , a root vertex \(r\in V\) , and a set \(K\subseteq V\setminus\{r\}\) of \(k\) terminals. The goal is to find a minimum-cost subgraph of \(G\) that contains a path from \(r\) to every terminal \(t\in K\) . DST has been a notorious problem for decades as there is a large gap between the best-known polynomial-time approximation ratio of \(O(k^{\epsilon})\) for any constant \(\epsilon {\gt} 0\) , and the best quasi-polynomial-time approximation ratio of \(O\left(\frac{\log^{2}k}{\log\log k}\right)\) . Towards understanding this gap, we study the integrality gap of the standard flow linear programming relaxation for the problem. We show that the linear program (LP) has an integrality gap of \(\Omega(n^{0.0418})\) . Previously, the integrality gap of the LP is only known to be \(\Omega\left(\frac{\log^{2}n}{\log\log n}\right)\) [Halperin et al., SODA’03 & SIAM J. Comput.] and \(\Omega(\sqrt{k})\) [Zosin-Khuller, SODA’02] in some instance with \(\sqrt{k}=O\left(\frac{\log n}{\log\log n}\right)\) . Our result gives the first known lower bound on the integrality gap of this standard LP that is polynomial in \(n\) , the number of vertices. Consequently, we rule out the possibility of developing a poly-logarithmic approximation algorithm for the problem based on the flow LP relaxation.

A novel branch‐and‐bound algorithm for solving linear multiplicative programming problems

AbstractThis article proposes a rectangular branch‐and‐bound algorithm for solving linear multiplication problems (LMP) globally. In order to obtain a reliable lower bound of the original problem, this article designs a novel linear relaxation programming problem (LRP) that has not been seen in the existing literature. Based on the basic framework of the rectangular branch and bound algorithm, this article proposes an algorithm that can obtain a global solution. According to the structure of linear relaxation programming, the article designs a region reduction technology to improve the efficiency of the algorithm. This article also provides convergence analysis to ensure the reliability of the algorithm. Finally, several numerical experiments are used to demonstrate the effectiveness and robustness of the algorithm.

Rank-One Boolean Tensor Factorization and the Multilinear Polytope

We consider the NP-hard problem of finding the closest rank-one binary tensor to a given binary tensor, which we refer to as the rank-one Boolean tensor factorization (BTF) problem. This optimization problem can be used to recover a planted rank-one tensor from noisy observations. We formulate rank-one BTF as the problem of minimizing a linear function over a highly structured multilinear set. Leveraging on our prior results regarding the facial structure of multilinear polytopes, we propose novel linear programming relaxations for rank-one BTF. We then establish deterministic sufficient conditions under which our proposed linear programs recover a planted rank-one tensor. To analyze the effectiveness of these deterministic conditions, we consider a semirandom model for the noisy tensor and obtain high probability recovery guarantees for the linear programs. Our theoretical results as well as numerical simulations indicate that certain facets of the multilinear polytope significantly improve the recovery properties of linear programming relaxations for rank-one BTF. Funding: A. Del Pia is partially funded by the Air Force Office of Scientific Research [Grant FA9550-23-1-0433]. A. Khajavirad is partially funded by the Air Force Office of Scientific Research [Grant FA9550-23-1-0123].

Use of Machine Learning Models to Warmstart Column Generation for Unit Commitment

The unit commitment problem is an important optimization problem in the energy industry used to compute the most economical operating schedules of power plants. Typically, this problem has to be solved repeatedly with different data but with the same problem structure. Machine learning techniques have been applied in this context to find primal feasible solutions. Dantzig-Wolfe decomposition with a column generation procedure is another approach that has been shown to be successful in solving the unit commitment problem to tight tolerance. We propose the use of machine learning models not to find primal feasible solutions directly but to generate initial dual values for the column generation procedure. Our numerical experiments compare machine learning–based methods for warmstarting the column generation procedure with three baselines: column prepopulation, the linear programming relaxation, and coldstart. The experiments reveal that the machine learning approaches are able to find both tight lower bounds and accurate primal feasible solutions in a shorter time compared with the baselines. Furthermore, these approaches scale well to handle large instances. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms—Discrete.

Optimal refinement of strata to balance covariates.

What is the best way to split one stratum into two to maximally reduce the within-stratum imbalance in many covariates? We formulate this as an integer program and approximate the solution by randomized rounding of a linear program. A linear program may assign a fraction of a person to each refined stratum. Randomized rounding views fractional people as probabilities, assigning intact people to strata using biased coins. Randomized rounding is a well-studied theoretical technique for approximating the optimal solution of certain insoluble integer programs. When the number of people in a stratum is large relative to the number of covariates, we prove the following new results: (i) randomized rounding to split a stratum does very little randomizing, so it closely resembles the linear programming relaxation without splitting intact people; (ii) the linear relaxation and the randomly rounded solution place lower and upper bounds on the unattainable integer programming solution; and because of (i), these bounds are often close, thereby ratifying the usable randomly rounded solution. We illustrate using an observational study that balanced many covariates by forming matched pairs composed of 2016 patients selected from 5735 using a propensity score. Instead, we form 5 propensity score strata and refine them into 10 strata, obtaining excellent covariate balance while retaining all patients. An R package optrefine at CRAN implements the method. Supplementary materials are available online.

Dynamic capacity allotment planning for airlines via travel agencies

In this paper, we study the dynamic capacity allocation problem of an airline using allotment contracts among travel agencies. The capacity allocation problem is formulated as a dynamic programming model where the initial fixed allotment is determined on the basis of a contract between agencies and the airline while the variable allotment is distributed over the planning horizon. We develop approximation methods based on linear programming and Lagrangian relaxation to solve the underlying dynamic programming model. The computational experiments are designed to illustrate the performance of the proposed approaches using real data. The numerical results show that the capacity allotment policy in terms of fixed and variable allotments is significantly affected by price and demand patterns. Even though fixed allotments can provide a cushion against the loss due to potential empty seats, they may not be in the best interest of the airline. Moreover, the booking rules and the airline’s capacity allocation strategy have an impact on the airline’s revenue and the agency’s profit.