Searcharxiv⌕ Search

arXiv subjects

Santanu S. Dey

Publications and source records attributed to Santanu S. Dey.

At least 37 records · Page 2Linked to original sources
arXiv2024

Branching with a pre-specified finite list of $k$-sparse split sets for binary MILPs

When branching for binary mixed integer linear programs with disjunctions of sparsity level $2$, we observe that there exists a finite list of $2$-sparse disjunctions, such that any other $2$-sparse disjunction is dominated by one disjunction in this finite list. For sparsity level greater than $2$, we show that a finite list of disjunctions with this property cannot exist. This leads to the definition of covering number for a list of splits disjunctions. Given a finite list of split sets $\mathcal{F}$ of $k$-sparsity, and a given $k$-sparse split set $S$, let $\mathcal{F}(S)$ be the minimum number of split sets from the list $\mathcal{F}$, whose union contains $S \cap [0, \ 1]^n$. Let the covering number of $\mathcal{F}$ be the maximum value of $\mathcal{F}(S)$ over all $k$-sparse split sets $S$. We show that the covering number for any finite list of $k$-sparse split sets is at least $\lfloor k/2\rfloor $ for $k \geq 4$. We also show that the covering number of the family of $k$-sparse split sets with coefficients in $\{-1, 0, 1\}$ is upper bounded by $k-1$ for $k \leq 4$.

math.OC↗
arXiv2024

Non-Monotonicity of Branching Rules with respect to Linear Relaxations

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 non-monotonic. 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 non-monotonicity in practice while using full strong branching. We consider randomly generated multi-dimensional 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 non-monotonicity. However, when a sufficiently large gap is closed, a significant decrease in tree size may be expected.

math.OC↗
arXiv2024

A reformulation-enumeration MINLP algorithm for gas network design

Gas networks are used to transport natural gas, which is an important resource for both residential and industrial customers throughout the world. The gas network design problem is generally modelled as a nonconvex mixed-integer nonlinear integer programming problem (MINLP). The challenges of solving the resulting MINLP arise due to the nonlinearity and nonconvexity. In this paper, we propose a framework to study the "design variant" of the problem in which the variables are the diameter choices of the pipes, the flows, the potentials, and the states of various network components. We utilize a nested loop that includes a two-stage procedure that involves a convex reformulation of the original problem in the inner loop and an efficient enumeration scheme in the outer loop. We conduct experiments on benchmark networks to validate and analyze the performance of our framework.

math.OC↗
arXiv2023

On Sparse Canonical Correlation Analysis

The classical Canonical Correlation Analysis (CCA) identifies the correlations between two sets of multivariate variables based on their covariance, which has been widely applied in diverse fields such as computer vision, natural language processing, and speech analysis. Despite its popularity, CCA can encounter challenges in explaining correlations between two variable sets within high-dimensional data contexts. Thus, this paper studies Sparse Canonical Correlation Analysis (SCCA) that enhances the interpretability of CCA. We first show that SCCA generalizes three well-known sparse optimization problems, sparse PCA, sparse SVD, and sparse regression, which are all classified as NP-hard problems. This result motivates us to develop strong formulations and efficient algorithms. Our main contributions include (i) the introduction of a combinatorial formulation that captures the essence of SCCA and allows the development of approximation algorithms; (ii) the derivation of an equivalent mixed-integer semidefinite programming model that facilitates a specialized branch-and-cut algorithm with analytical cuts; and (iii) the establishment of the complexity results for two low-rank special cases of SCCA. The effectiveness of our proposed formulations and algorithms is validated through numerical experiments.

math.OC↗
arXiv2023

Aggregations of quadratic inequalities and hidden hyperplane convexity

We study properties of the convex hull of a set $S$ described by quadratic inequalities. A simple way of generating inequalities valid on $S$ is to take a nonnegative linear combinations of the defining inequalities of $S$. We call such inequalities aggregations. Special aggregations naturally contain the convex hull of $S$, and we give sufficient conditions for such aggregations to define the convex hull. We introduce the notion of hidden hyperplane convexity (HHC), which is related to the classical notion of hidden convexity of quadratic maps. We show that if the quadratic map associated with $S$ satisfies HHC, then the convex hull of $S$ is defined by special aggregations. To the best of our knowledge, this result generalizes all known results regarding aggregations defining convex hulls. Using this sufficient condition, we are able to recognize previously unknown classes of sets where aggregations lead to convex hull. We show that the condition known as positive definite linear combination together with hidden hyerplane convexity is a sufficient condition for finitely many aggregations to define the convex hull. All the above results are for sets defined using open quadratic inequalities. For closed quadratic inequalities, we prove a new result regarding aggregations giving the convex hull, without topological assumptions on $S$.

math.OC↗
arXiv2022

Exploiting Instance and Variable Similarity to Improve Learning-Enhanced Branching

In many operational applications, it is necessary to routinely find, within a very limited time window, provably good solutions to challenging mixed-integer linear programming (MILP) problems. An example is the Security-Constrained Unit Commitment (SCUC) problem, solved daily to clear the day-ahead electricity markets. Previous research demonstrated that machine learning (ML) methods can produce high-quality heuristic solutions to combinatorial problems, but proving the optimality of these solutions, even with recently-proposed learning-enhanced branching methods, can still be time-consuming. In this paper, we propose a simple modification to improve the performance of learning-enhanced branching methods based on the key observation that, in such operational applications, instances are significantly similar to each other. Specifically, instances typically share the same size and problem structure, with slight differences only on matrix coefficients, right-hand sides and objective function. In addition, certain groups of variables within a given instance are also typically similar to each other. Therefore, unlike previous works in the literature which predicted all branching scores with a single ML model, we propose training separate ML models per variable or per groups of variables, based on their similarity. We evaluate this enhancement on realistic large-scale SCUC instances and we obtain significantly better gap closures than previous works with the same amount of training data.

math.OC↗
arXiv2022

Solving sparse separable bilinear programs using lifted bilinear cover inequalities

Recently, we proposed a class of inequalities called lifted bilinear cover inequalities, which are second-order cone representable convex inequalities, and 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 semi-definite 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 to when they are not.

math.OC↗
arXiv2022

Solving sparse principal component analysis with global support

Sparse principal component analysis with global support (SPCAgs), is the problem of finding the top-$r$ leading principal components such that all these principal components are linear combinations of a common subset of at most $k$ variables. SPCAgs is a popular dimension reduction tool in statistics that enhances interpretability compared to regular principal component analysis (PCA). Methods for solving SPCAgs in the literature are either greedy heuristics (in the special case of $r = 1$) with guarantees under restrictive statistical models or algorithms with stationary point convergence for some regularized reformulation of SPCAgs. Crucially, none of the existing computational methods can efficiently guarantee the quality of the solutions obtained by comparing them against dual bounds. In this work, we first propose a convex relaxation based on operator norms that provably approximates the feasible region of SPCAgs within a $c_1 + c_2 \sqrt{\log r} = O(\sqrt{\log r})$ factor for some constants $c_1, c_2$. To prove this result, we use a novel random sparsification procedure that uses the Pietsch-Grothendieck factorization theorem and may be of independent interest. We also propose a simpler relaxation that is second-order cone representable and gives a $(2\sqrt{r})$-approximation for the feasible region. Using these relaxations, we then propose a convex integer program that provides a dual bound for the optimal value of SPCAgs. Moreover, it also has worst-case guarantees: it is within a multiplicative/additive factor of the original optimal value, and the multiplicative factor is $O(\log r)$ or $O(r)$ depending on the relaxation used. Finally, we conduct computational experiments that show that our convex integer program provides, within a reasonable time, good upper bounds that are typically significantly better than the natural baselines.

math.OC↗
arXiv2022

Lower Bounds on the Size of General Branch-and-Bound Trees

A \emph{general branch-and-bound tree} is a branch-and-bound tree which is allowed to use general disjunctions of the form $π^{\top} x \leq π_0 \,\vee\, π^{\top}x \geq π_0 + 1$, where $π$ is an integer vector and $π_0$ is an integer scalar, to create child nodes. We construct a packing instance, a set covering instance, and a Traveling Salesman Problem instance, such that any general branch-and-bound tree that solves these instances must be of exponential size. We also verify that an exponential lower bound on the size of general branch-and-bound trees persists when we add Gaussian noise to the coefficients of the cross polytope, thus showing that polynomial-size "smoothed analysis" upper bound is not possible. The results in this paper can be viewed as the branch-and-bound analog of the seminal paper by Chvátal et al. \cite{chvatal1989cutting}, who proved lower bounds for the Chvátal-Gomory rank.

math.OC↗
arXiv2021

Node-based valid inequalities for the optimal transmission switching problem

The benefits of transmission line switching are well-known in terms of reducing operational cost and improving system reliability of power systems. However, finding the optimal power network configuration is a challenging task due to the combinatorial nature of the underlying optimization problem. In this work, we identify a certain "node-based" set that appears as substructure of the optimal transmission switching problem and then conduct a polyhedral study of this set. We construct an extended formulation of the integer hull of this set and present the inequality description of the integer hull in the original space in some cases. These inequalities in the original space can be used as cutting-planes for the transmission line switching problem. Finally, we present the results of our computational experiments using these cutting-planes on difficult test cases from the literature.

math.OC↗
arXiv2021

A Theoretical and Computational Analysis of Full Strong-Branching

Full strong-branching is a well-known variable selection rule that is known experimentally to produce significantly smaller branch-and-bound trees in comparison to all other known variable selection rules. In this paper, we attempt an analysis of the performance of the strong-branching rule both from a theoretical and a computational perspective. On the positive side for strong-branching we identify vertex cover as a class of instances where this rule provably works well. In particular, for vertex cover we present an upper bound on the size of the branch-and-bound tree using strong-branching as a function of the additive integrality gap, show how the Nemhauser-Trotter property of persistency which can be used as a pre-solve technique for vertex cover is being recursively and consistently used throughout the strong-branching based branch-and-bound tree, and finally provide an example of a vertex cover instance where not using strong-branching leads to a tree that has at least exponentially more nodes than the branch-and-bound tree based on strong-branching. On the negative side for strong-branching, we identify another class of instances where strong-branching based branch-and-bound tree has exponentially larger tree in comparison to another branch-and-bound tree for solving these instances. On the computational side, we conduct experiments on various types of instances to understand how much larger is the size of the strong-branching based branch-and-bound tree in comparison to the optimal branch-and-bound tree. The main take-away from these experiments is that for all these instances, the size of the strong-branching based branch-and-bound tree is within a factor of two of the size of the optimal branch-and-bound tree.

math.OC↗
arXiv2021

Branch-and-Bound Solves Random Binary IPs in Polytime

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.

math.OC↗
arXiv2021

A Scalable Lower Bound for the Worst-Case Relay Attack Problem on the Transmission Grid

We consider a bilevel attacker-defender problem to find the worst-case attack on the relays that control the transmission grid. The attacker maximizes load shed by infiltrating a number of relays and rendering the components connected to them inoperable. The defender responds by minimizing the load shed, re-dispatching using a DC optimal power flow (DCOPF) problem on the remaining network. Though worst-case interdiction problems on the transmission grid are well-studied, 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 making the problem 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, where 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, 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 1, so we can solve our restriction exactly. Last, we see empirically that this formulation scales well computationally. Through experiments on 16 networks with up to 6468 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 addition, calculating the bound is approximately 150 times faster than achieving the same bound with the reformulation guessing the dual bounds.

math.OC↗
arXiv2021

Using L1-relaxation and integer programming to obtain dual bounds for sparse PCA

Principal component analysis (PCA) is one of the most widely used dimensionality reduction tools in data analysis. The PCA direction is a linear combination of all features with nonzero loadings -- this impedes interpretability. Sparse PCA (SPCA) is a framework that enhances interpretability by incorporating an additional sparsity requirement in the feature weights. However, unlike PCA, the SPCA problem is NP-hard. Most conventional methods for solving SPCA are heuristics with no guarantees, such as certificates of optimality on the solution-quality via associated dual bounds. Dual bounds are available via standard semidefinite programming (SDP) based relaxations, which may not be tight, and the SDPs are difficult to scale by off-the-shelf solvers. In this paper, we present a convex integer programming (IP) framework to derive dual bounds. At the heart of our approach is the so-called $\ell_1$-relaxation of SPCA. While the $\ell_1$-relaxation leads to convex optimization problems for $\ell_0$-sparse linear regression and relatives, it results in a non-convex optimization problem for the PCA problem. We first show that the $\ell_1$-relaxation gives a tight multiplicative bound on SPCA. Then we show how to use standard integer programming techniques to further relax the $\ell_1$-relaxation into a convex IP. We present worst-case results on the quality of the dual bound from the convex IP. We observe that the dual bounds are significantly better than worst-case performance and are superior to the SDP bounds in some real-life instances. Moreover, solving the convex IP model using commercial IP solvers appears to scale much better than solving the SDP-relaxation using commercial solvers. To the best of our knowledge, we obtain the best dual bounds for real and artificial instances for SPCA problems involving covariance matrices of size up to $2000\times 2000$.

math.OC↗
arXiv2021

Hyperbolic Relaxation of $k$-Locally Positive Semidefinite Matrices

A successful computational approach for solving large-scale positive semidefinite (PSD) programs is to enforce PSD-ness on only a collection of submatrices. For our study, we let $\mathcal{S}^{n,k}$ be the convex cone of $n\times n$ symmetric matrices where all $k\times k$ principal submatrices are PSD. We call a matrix in this $k$-\emph{locally PSD}. In order to compare $S^{n,k}$ to the of PSD matrices, we study eigenvalues of $k$-{locally PSD} matrices. The key insight in this paper is that there is a convex cone $H(e_k^n)$ so that if $X \in \mathcal{S}^{n,k}$, then the vector of eigenvalues of $X$ is contained in $H(e_k^n)$. The cone $H(e_k^n)$ is the hyperbolicity cone of the elementary symmetric polynomial $e^k_n$ (where $e_k^n(x) = \sum_{S \subseteq [n] : |S| = k} \prod_{i \in S} x_i$) with respect to the all ones vector. Using this insight, we are able to improve previously known upper bounds on the Frobenius distance between matrices in $\mathcal{S}^{n,k}$ and PSD matrices. We also study the quality of the convex relaxation $H(e^n_k)$. We first show that this relaxation is tight for the case of $k = n -1$, that is, for every vector in $H(e^n_{n -1})$ there exists a matrix in $\mathcal{S}^{n, n -1}$ whose eigenvalues are equal to the components of the vector. We then prove a structure theorem on nonsingular matrices in $\mathcal{S}^{n,k}$ all of whose $k\times k$ principal minors are zero, which we believe is of independent interest. %We then prove a structure theorem that precisely characterizes the non-singular matrices in $\mathcal{S}^{n,k}$ whose vector of eigenvalues belongs to the boundary of $H(e^n_k)$. This result shows shows that for $1< k < n -1$ "large parts" of the boundary of $H(e_k^n)$ do not intersect with the eigenvalues of matrices in $\mathcal{S}^{n,k}$.

math.OC↗
arXiv2021

Lifting convex inequalities for bipartite bilinear programs

The goal of this paper is to derive new classes of valid convex inequalities for quadratically constrained quadratic programs (QCQPs) through the technique of lifting. Our first main result shows that, for sets described by one bipartite bilinear constraint together with bounds, it is always possible to sequentially lift a seed inequality that is valid for a restriction obtained by fixing variables to their bounds, when the lifting is accomplished using affine functions of the fixed variables. In this setting, sequential lifting involves solving a non-convex nonlinear optimization problem each time a variable is lifted, just as in Mixed Integer Linear Programming. To reduce the computational burden associated with this procedure, we develop a framework based on subadditive approximations of lifting functions that permits sequence-independent lifting of seed inequalities for separable bipartite bilinear sets. In particular, this framework permits the derivation of closed-form valid inequalities. We then study a separable bipartite bilinear set where the coefficients form a minimal cover with respect to the right-hand-side. For this set, we introduce a bilinear cover inequality, which is second-order cone representable. We argue that this bilinear cover inequality is strong by showing that it yields a constant-factor approximation of the convex hull of the original set. We study its lifting function and construct a two-slope subadditive upper bound. Using this subadditive approximation, we lift fixed variable pairs in closed-form, thus deriving a lifted bilinear cover inequality that is valid for general separable bipartite bilinear sets with box constraints.

math.OC↗
arXiv2021

On obtaining the convex hull of quadratic inequalities via aggregations

A classical approach for obtaining valid inequalities for a set involves weighted aggregations of the inequalities that describe such set. When the set is described by linear inequalities, thanks to the Farkas lemma, we know that every valid inequality can be obtained using aggregations. When the inequalities describing the set are two quadratics, Yildiran showed that the convex hull of the set is given by at most two aggregated inequalities. In this work, we study the case of a set described by three or more quadratic inequalities. We show that, under technical assumptions, the convex hull of a set described by three quadratic inequalities can be obtained via (potentially infinitely many) aggregated inequalities. We also show, through counterexamples, that it is unlikely to have a similar result if either the technical conditions are relaxed, or if we consider four or more inequalities.

math.OC↗