In modeling battery energy storage systems (BESS) in power systems, binary variables are used to represent the complementary nature of charging and discharging. A conventional approach for these BESS optimization problems is to relax binary variables and convert the problem into a linear program. However, such linear programming relaxation models can yield unrealistic fractional solutions, such as simultaneous charging and discharging. In this paper, we develop a regularized mixed-integer programming (MIP) model for the optimal power flow (OPF) problem with BESS. We prove that, under mild conditions, the proposed regularized model admits a zero integrality gap with its linear programming relaxation; hence, it can be solved efficiently. By studying the properties of the regularized MIP model, we show that its optimal solution is also near optimal to the original OPF problem with BESS, thereby providing a valid and tight upper bound for the OPF problem with BESS. The use of the regularized MIP model allows us to solve a trilevel [Formula: see text]-[Formula: see text]-[Formula: see text] network contingency problem, which is otherwise intractable to solve. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms–Discrete. Funding: N. Jiang (as a graduate student at the Georgia Institute of Technology) and W. Xie were supported in part by the National Science Foundation [Grant 2246414] and the Office of Naval Research [Grant N00014-24-1-2066]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2024.0771 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2024.0771 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Modern mixed-integer programming solvers use the branch-and-cut framework, where cutting planes are added to improve the tightness of the linear programming (LP) relaxation, with the expectation that the tighter formulation would produce smaller branch-and-bound trees. In this work, we consider the question of whether adding cuts will always lead to smaller trees for a given fixed branching rule. We formally call such a property of a branching rule monotonicity. We prove that any branching rule which exclusively branches on fractional variables in the LP solution is nonmonotonic. Moreover, we present a family of instances where adding a single cut leads to an exponential increase in the size of full strong branching trees, despite improving the LP bound. Finally, we empirically attempt to estimate the prevalence of nonmonotonicity in practice while using full strong branching. We consider randomly generated multidimensional knapsacks tightened by cover cuts as well as instances from the MIPLIB 2017 benchmark set for the computational experiments. Our main insight from these experiments is that if the gap closed by cuts is small, change in tree size is difficult to predict, and often increases, possibly due to inherent nonmonotonicity. However, when a sufficiently large gap is closed, a significant decrease in tree size may be expected. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms—Discrete. Funding: This work was supported by Air Force Office of Scientific Research [Grant F9550-22-1-0052]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2024.0709 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2024.0709 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Abstract In this paper, we study the strength of convex relaxations obtained by convexification of aggregation of constraints for a set S described by two bilinear bipartite equalities. Aggregation is the process of rescaling the original constraints by scalar weights and adding the scaled constraints together. It is natural to study the aggregation technique as it yields a single bilinear bipartite equality whose convex hull is already understood from previous literature. On the theoretical side, we present sufficient conditions when $$\text {conv} (S)$$ conv ( S ) can be described by the intersection of convex hulls of a finite number of aggregations, examples when $$\text {conv} (S)$$ conv ( S ) can only be obtained as the intersection of the convex hull of an infinite number of aggregations, and examples when $$\text {conv} (S)$$ conv ( S ) cannot be achieved exactly from the process of aggregation. Computationally, we explore different methods to derive aggregation weights in order to obtain tight convex relaxations. We show that even if an exact convex hull may not be achieved using aggregations, including the convex hull of an aggregation often significantly tightens the outer approximation of $$\text {conv} (S)$$ conv ( S ) . Finally, we apply the aggregation method to obtain convex relaxation for the structural model updating problem and show that this yields better bounds within a branch-and-bound tree as compared to not using aggregations.

Recently a class of second-order cone representable convex inequalities called lifted bilinear cover inequalities were introduced, which are valid for a set described by a separable bilinear constraint together with bounds on variables. In this paper, we study the computational potential of these inequalities for separable bilinear optimization problems. We first prove that the semidefinite programming relaxation provides no benefit over the McCormick relaxation for such problems. We then design a simple randomized separation heuristic for lifted bilinear cover inequalities. In our computational experiments, we separate many rounds of these inequalities starting from McCormick’s relaxation of instances where each constraint is a separable bilinear constraint set. We demonstrate that there is a significant improvement in the performance of a state-of-the-art global solver in terms of gap closed, when these inequalities are added at the root node compared with when they are not. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms – Discrete. Funding: S. S. Dey gratefully acknowledges the support by the Office of Naval Research [Grant N000141912323].

One of the most complicating factors in decentralized solution methods for a broad range of power system optimization problems is the modeling of power flow equations. Existing formulations for direct current power flows either have limited scalability or are very dense and unstructured, making them unsuitable for large-scale decentralized studies. In this work, we present a novel sparsified variant of the injection shift factors formulation, which has a decomposable block-diagonal structure and scales well for large systems. We also propose a decentralized solution method, based on the alternating direction multiplier method, that efficiently handles transmission line outages in N-1 security requirements. Benchmarks on multizonal security-constrained unit commitment problems show that the proposed formulation and algorithm can reliably and efficiently solve interconnection-level test systems with up to 6,515 buses with no convergence or numerical issues. History: Accepted by Pascal Van Hentenryck, Area Editor for Computational Modeling: Methods & Analysis. Funding: This work was partially supported by Laboratory Directed Research and Development funding from Argonne National Laboratory provided by the Director, Office of Science, of the U.S. Department of Energy [Grant DE-AC02-06CH11357]. This work was also partially supported by the U.S. Department of Energy Advanced Grid Modeling Program [Grant DE-OE0000875]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2022.0326 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2022.0326 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

We consider a bilevel attacker–defender problem to find the worst-case attack on the relays that control transmission grid components. The attacker infiltrates some number of relays and renders all of the components connected to them inoperable with the goal of maximizing load shed. The defender responds by minimizing the resulting load shed, redispatching using a DC optimal power flow (DCOPF) problem on the remaining network. Though worst-case interdiction problems on the transmission grid have been studied for years, there remains a need for exact and scalable methods. Methods based on using duality on the inner problem rely on the bounds of the dual variables of the defender problem in order to reformulate the bilevel problem as a mixed integer linear problem. Valid dual bounds tend to be large, resulting in weak linear programming relaxations and, hence, making the problem more difficult to solve at scale. Often smaller heuristic bounds are used, resulting in a lower bound. In this work, we also consider a lower bound, but instead of bounding the dual variables, we drop the constraints corresponding to Ohm’s law, relaxing DCOPF to capacitated network flow. We present theoretical results showing that, for uncongested networks, approximating DCOPF with network flow yields the same set of injections and, thus, the same load shed, which suggests that this restriction likely gives a high-quality lower bound in the uncongested case. Furthermore, we show that, in the network flow relaxation of the defender problem, the duals are bounded by one, so we can solve our restriction exactly. Finally, because the big-M values in the linearization are equal to one and network flow has a well-known structure, we see empirically that this formulation scales well computationally with increased network size. Through empirical experiments on 16 networks with up to 6,468 buses, we find that this bound is almost always as tight as we can get from guessing the dual bounds even for congested networks in which the theoretical results do not hold. In addition, calculating the bound is approximately 150 times faster than achieving the same bound with the reformulation guessing the dual bounds.

Branch-and-bound is the workhorse of all state-of-the-art mixed integer linear programming (MILP) solvers. These implementations of branch-and-bound typically use variable branching, that is, the child nodes are obtained by fixing some variable to an integer value v in one node and to v+1 in the other node. Even though modern MILP solvers are able to solve very large-scale instances efficiently, relatively little attention has been given to understanding why the underlying branch-and-bound algorithm performs so well. In this paper our goal is to theoretically analyze the performance of the standard variable branching based branch-and-bound algorithm. In order to avoid the exponential worst-case lower bounds, we follow the common idea of considering random instances. More precisely, we consider random integer programs where the entries of the coefficient matrix and the objective function are randomly sampled. Our main result is that with good probability branch-and-bound with variable branching explores only a polynomial number of nodes to solve these instances, for a fixed number of constraints. To the best of our knowledge this is the first known such result for a standard version of branch-and-bound. We believe that this result provides a compelling indication of why branch-and-bound with variable branching works so well in practice.

Sparse principal component analysis (PCA) is a widely-used dimensionality reduction tool in statistics and machine learning. Most methods mentioned in literature are either heuristics for good primal feasible solutions under statistical assumptions or ADMM-type algorithms with stationary/critical points convergence property for the regularized reformulation of sparse PCA. However, none of these methods can efficiently verify the quality of the solutions via comparing current objective values with their dual bounds, especially in model-free case. We propose a new framework that finds out upper (dual) bounds for the sparse PCA within polynomial time via solving a convex integer program (IP). We show that, in the worst-case, the dual bounds provided by the convex IP is within an affine function of the global optimal value. Moreover, in contrast to the semi-definition relaxation, this framework is much easier to scale on large cases. Numerical results on both artificial and real cases are reported to demonstrate the advantages of our method.

Scenario decomposition algorithms for stochastic programs compute bounds by dualizing all nonanticipativity constraints and solving individual scenario problems independently. We develop an approach that improves on these bounds by reinforcing a carefully chosen subset of nonanticipativity constraints, effectively placing scenarios into groups. Specifically, we formulate an optimization problem for grouping scenarios that aims to improve the bound by optimizing a proxy metric based on information obtained from evaluating a subset of candidate feasible solutions. We show that the proposed grouping problem is NP-hard in general, identify a polynomially solvable case, and present two formulations for solving the problem: a matching formulation for a special case and a mixed-integer programming formulation for the general case. We use the proposed grouping scheme as a preprocessing step for a particular scenario decomposition algorithm and demonstrate its effectiveness in solving standard test instances of two-stage 0–1 stochastic programs. Using this approach, we are able to prove optimality for all previously unsolved instances of a standard test set. Additionally, we implement this scheme as a preprocessing step for PySP, a publicly available and widely used implementation of progressive hedging, and compare this grouping approach with standard grouping approaches on large-scale stochastic unit commitment instances. Finally, the idea is extended to propose a finitely convergent algorithm for two-stage stochastic programs with a finite feasible region.

We explore the lifting question in the context of cut-generating functions. Most of the prior literature on this question focuses on cut-generating functions that have the unique lifting property. We develop a general theory for understanding the lifting question for cut-generating functions that do not necessarily have the unique lifting property.

We consider the integer L-shaped method for two-stage stochastic integer programs. To improve the performance of the algorithm, we present and combine two strategies. First, to avoid time-consuming exact evaluations of the second-stage cost function, we propose a simple modification that alternates between linear and mixed-integer subproblems. Next, to better approximate the shape of the second-stage cost function, we present a general framework to generate optimality cuts via a cut-generating linear program that considers information from all solutions found up to any given stage of the method. To address the impact of the proposed approaches, we report computational results on two classes of stochastic integer problems.

Let ${{\textbf{C}}}$ be a full-dimensional pointed closed convex cone in ${\mathbb{R}}^m$ obtained by taking the conic hull of a strictly convex set. Given $A \in {\mathbb{Q}} ^{m \times n_1}$, $B \in \mathbb{Q}^{m \times n_2}$, and $b \in {\mathbb{Q}}^m$, a simple conic mixed-integer set (SCMIS) is a set of the form $\{(x,y)\in {\mathbb{Z}}^{n_1} \times {\mathbb{R}}^{n_2}\,|\,\ Ax +By -b \in {{\textbf{C}}}\}$. In this paper, we give a complete characterization of the closedness of convex hulls of SCMISs. Under certain technical conditions on the cone ${{\textbf{C}}}$, we show that the closedness characterization can be used to construct a polynomial-time algorithm to check the closedness of convex hulls of SCMISs. Moreover, we also show that the Lorentz cone satisfies these technical conditions. In the special case of pure integer problems, we present sufficient conditions, which can be checked in polynomial time, to verify the closedness of intersection of SCMISs.

Following the flurry of recent theoretical work on cutting planes from two-row mixed integer group relaxations of a linear programming tableau, we report on computational tests to evaluate the strength of two-row cuts based on lattice-free triangles having more than one integer point on one side. A heuristic procedure to generate such triangles (referred to in the literature as “type 2” triangles) is presented, and then the coefficients of the integer variables are tightened by lifting. To test the effectiveness of triangle cuts, we compare the gap closed using Gomory mixed integer cuts for one round, the gap closed in one round using all the triangle cuts generated by our heuristic, and the gap closed by a small number of two-row split cuts. Our tests are carried out on randomly generated instances designed to represent different problem features by varying the number of integer nonbasic variables, bounds, nonnegativity constraints, and density, as well as on the classical MIPLIB instances. The outcome of this computational analysis is some insight into key characteristics of MIP instances whose presence makes two-row triangle cuts computationally effective. In particular, it appears to be necessary that the tableau row pairs are dense, and more subjectively that the nonbasic continuous variables are “important.” Unfortunately these characteristics seem to be rarely present among real-life instances, and more specifically the tableau rows of the MIPLIB instances are far from dense.

We consider a class of linear programs involving a set of covering constraints of which at most k are allowed to be violated. We show that this covering linear program with violation is strongly 𝒩𝒫-hard. To improve the performance of mixed-integer programming-based schemes for these problems, we introduce and analyze a coefficient strengthening scheme, adapt and analyze an existing cutting plane technique, and present a branching technique. Through computational experiments, we empirically verify that these techniques are significantly effective in improving solution times over the CPLEX mixed-integer programming solver. In particular, we observe that the proposed schemes can cut down solution times from as much as six days to under four hours.

Let $S\subseteq\mathbb{Z}^n$ satisfy the property that $\mathrm{conv}(S)\cap\mathbb{Z}^n=S$. Then a convex set K is called an S-free convex set if $\mathrm{int}(K)\cap S=\emptyset$. A maximal S-free convex set is an S-free convex set that is not properly contained in any S-free convex set. We show that maximal S-free convex sets are polyhedra. This result generalizes a result of Basu et al. [SIAM J. Discrete Math., 24 (2010), pp. 158–168] for the case where S is the set of integer points in a rational polyhedron and a result of Lovász [Mathematical Programming: Recent Developments and Applications, M. Iri and K. Tanabe, eds., Kluwer, Dordrecht, 1989, pp. 177–210] and Basu et al. [Math. Oper. Res., 35 (2010), pp. 704–720] for the case where S is the set of integer points in some affine subspace of $\mathbb{R}^n$.

We propose an approximate lifting procedure for general integer programs. This lifting procedure uses information from multiple constraints of the problem formulation and can be used to strengthen formulations and cuts for mixed-integer programs. In particular, we demonstrate how it can be applied to improve Gomory's fractional cut, which is central to Glover's primal cutting-plane algorithm. We show that the resulting algorithm is finitely convergent. We also present numerical results that illustrate the computational benefits of the proposed lifting procedure.