arXiv subjects

Santanu S. Dey

Publications and source records attributed to Santanu S. Dey.

At least 19 recordsLinked to original sources

arXiv2026

Decomposition of Sparse Integer Programs via Nonlinear Edge Encodings and Column-and-Row Generation

A wide range of sparse integer programs admit a block structure in which subproblems interact through a small set of shared variables. Dualizing the linking equalities yields a decomposable Lagrangian relaxation, but generally introduces a duality gap. Recent work shows that this gap can be closed while preserving decomposability by dualizing exponentially large families of redundant nonlinear consistency constraints on the shared variables. We develop a computational framework for exploiting this idea without explicitly constructing the resulting exponentially large relaxation. Our framework combines nonlinear edge encodings of shared-variable consistency with a column-and-row generation (CRG) algorithm that generates local integer solutions by pricing and encoding constraints by separation. With complete encodings and exact separation, the framework recovers the exact relaxation while retaining independent optimization over the blocks. We introduce several encoding families and establish exponential separations among them: the Generalized family can require exponentially fewer constraints than the Vertex, Monomial, or Reflected families, yet can itself require exponentially many constraints on instances for which a single problem-specific encoding suffices. Computational experiments on decomposed stable-set and dominating-set instances show that CRG substantially outperforms a monolithic formulation across a range of tree topologies and coupling strengths. The results also show that richer encoding families need not perform better computationally, highlighting the choice of encoding as a central issue in effective decomposition.

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arXiv2026

Change-Point Detection via Piecewise Linear Fitting Using MIP

We present a new mixed-integer programming (MIP) approach for offline multiple change-point detection by casting the problem as a globally optimal piecewise linear (PWL) fitting problem. Our main contribution is a family of strengthened MIP formulations whose linear programming (LP) relaxations admit integral projections onto the segment-assignment variables, which encode the segment membership of each data point. This property yields provably tighter relaxations than existing formulations for offline multiple change-point detection. We further extend the framework to multi-dimensional PWL models with shared change-points. Extensive computational experiments on benchmark real-world datasets demonstrate that the proposed formulations achieve reductions in solution times in comparison to the state-of-the-art.

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arXiv2026

Copositive Matrices with Ordered Off-Diagonal Entries

We study copositive matrices which admit a decomposition into a sum of a positive semidefinite matrix and a matrix with nonnegative entries. Our main result shows that if the off-diagonal entries of a copositive matrix are nondecreasing in rows and in columns, then it admits such a decomposition. We apply this result to study optimization of quadratic forms over the standard simplex. As a corollary, we obtain that a natural relaxation of this problem is tight when the objective function is separable, resolving an open question of Dey and Kocuk.

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arXiv2026

Probabilistic analysis of dual decomposition on two-stage stochastic integer programs

Two-stage stochastic integer programs provide a powerful framework for modeling decision-making under uncertainty, but they are notoriously difficult to solve at scale due to their high dimensionality and intrinsic nonconvexity. Decomposition-based algorithms such as Benders methods and Branch-and-Price (related dual decomposition methods) have become standard computational approaches for such problems and demonstrate excellent empirical performance in practice. Despite their widespread use, however, existing theoretical guarantees are almost exclusively based on worst-case analyses, which predict exponential convergence behavior in the problem dimension and fail to explain the strong performance observed in practice. In this paper, we present the first average-case analysis of Branch-and-Price for a broad class of two-stage stochastic binary integer programs. We study a stochastic-input model in which objective coefficients and constraint matrices are drawn at random and right-hand-side vectors scale with the decision dimension, while the number of constraints per scenario is fixed. Under this model, we prove that, with high probability, Branch-and-Price explores at most n^O(log s)nodes, yielding a quasi-polynomial bound on the size of the search tree in typical instances, where n denotes the decision dimension and s the number of scenarios. A key ingredient of our analysis is an average-case bound on the integrality gap of the natural linear programming (LP) relaxation. We show that this gap shrinks at rate O((logs log^2 n)/n)with high probability. This result is of independent interest, as it implies that the integrality gap grows only logarithmically with the number of scenarios on average.

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arXiv2026

A second-order cone representable class of nonconvex quadratic programs

We consider the problem of minimizing a sparse nonconvex quadratic function over the unit hypercube. By developing an extension of the Reformulation-Linearization Technique (RLT) to continuous quadratic sets, we propose a novel second-order cone (SOC) representable relaxation for this problem. By exploiting the sparsity of the quadratic function, we establish a sufficient condition under which the convex hull of the feasible region of the lifted quadratic program is SOC-representable. While the proposed formulation may be of exponential size in general, we identify additional structural conditions that guarantee the existence of a polynomial-size SOC-representable formulation, which can be constructed in polynomial time. Under these conditions, the optimal value of the nonconvex quadratic program coincides with that of a polynomial-size second-order cone program. Our results serve as a starting point for bridging the gap between the Boolean quadric polytope of sparse problems and its continuous counterpart.

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arXiv2026

Accelerating Column Generation in Highly Degenerate Integer Programming Problems with Template Pricing

We propose a new pricing strategy for column generation (CG), referred to as Template pricing. This method is motivated by the desire to coordinate solutions of different pricing subproblems in order to accelerate the convergence of the CG process and simultaneously obtain good quality integer feasible solutions. Instead of finding a column with the optimal reduced cost, Template pricing tries to maximize the similarity of columns with a given template vector, while restricting the search to columns with suitable reduced cost. We present an exact and heuristic method (based on Lagrangian relaxation) to efficiently solve the Template pricing problem. We conduct extensive computational experiments on benchmark instances of the Generalized Assignment Problem (GAP). Our results demonstrate that Template pricing can significantly accelerate the CG algorithm, especially in the presence of significant degeneracy, where several benchmark GAP instances solved over 1000x faster than Dantzig pricing, and over 100x with adaptive dual-smoothing. Template pricing allows us to achieve CG optimal bounds on all 1735 ISA instances, finding stronger bounds in 43% and improved integer solutions in 9% of these instances than previously released.

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arXiv2026

Chvátal-Gomory Rounding of Eigenvector Inequalities for QCQPs

We introduce and analyze a class of valid inequalities for nonconvex quadratically constrained optimization problems (QCQPs) which we call Eigen-CG inequalities. These inequalities are obtained by applying a Chvátal-Gomory (CG) rounding to the well-known eigenvector inequalities for QCQPs, and transferring binary-valid inequalities to the continuous setting via a result of Burer and Letchford (2009). We define three nested subfamilies and prove that they are strictly contained in one another. However, we show that the convex conic closure of two of these subfamilies is equal and, in fact, coincides with the Boros-Hammer inequalities -- a powerful family of inequalities that include, in particular, the triangle and McCormick inequalities. Using this CG perspective, we also prove that dense Eigen-CG inequalities are ineffective when used with the standard SDP+McCormick relaxation. This provides a complementary perspective on what is observed in practice: that sparse inequalities are impactful. Finally, based on these insights, we develop a computational strategy to find sparse Eigen-CG cuts and verify their effectiveness in nonconvex QCQP instances. Our results confirm that density quickly degrades effectiveness, but that including sparse inequalities beyond triangle inequalities can provide significant improvements in dual bounds.

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arXiv2025

Strengthening Dual Bounds for Multicommodity Capacitated Network Design with Unsplittable Flow Constraints

Multicommodity capacitated network design (MCND) models can be used to optimize the consolidation of shipments within e-commerce fulfillment networks. In practice, fulfillment networks require that shipments with the same origin and destination follow the same transfer path. This unsplittable flow requirement complicates the MCND problem, requiring integer programming (IP) formulations in which binary variables replace continuous flow variables. To enhance the solvability of this variant of the MCND problem for large-scale logistics networks, this work focuses on strengthening dual bounds. We investigate the polyhedra of arc-set relaxations, and we introduce two new classes of valid inequalities that can be implemented within solution approaches. We develop one approach that dynamically adds valid inequalities to the root node of a reformulation of the MCND IP with additional valid metric inequalities. We show the effectiveness of our ideas with a comprehensive computational study using path-based fulfillment instances, constructed from data provided by a large U.S.-based e-commerce company, and the well-known arc-based Canad instances. Experiments show that our best solution approach for a practical path-based model reduces the IP gap by an average of 26.5% and 22.5% for the two largest instance groups, compared to solving the reformulation alone, demonstrating its effectiveness in improving the dual bound. In addition, experiments using only the arc-based relaxation highlight the strength of our new valid inequalities relative to the linear programming relaxation (LPR), yielding an IP-gap reduction of more than 85%.

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arXiv2025

Extreme Strong Branching for QCQPs

For mixed-integer programs (MIPs), strong branching is a highly effective variable selection method to reduce the number of nodes in the branch-and-bound algorithm. Extending it to nonlinear problems is conceptually simple but practically limited. Branching on a binary variable fixes the variable to 0 or 1, whereas branching on a continuous variable requires an additional decision to choose a branching point. Previous extensions of strong branching predefine this point and then solve $2n$ relaxations where $n$ is the number of candidate variables to branch. We propose extreme strong branching, which evaluates multiple branching points per variable and jointly selects both the branching variable and point based on the objective value improvement. This approach resembles the success of strong branching for MIPs while additionally exploiting bound tightening as a byproduct. For certain types of quadratically constrained quadratic programs (QCQPs), computational experiments show that the extreme strong branching rule outperforms existing commercial solvers.

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arXiv2025

Convexification of a Separable Function over a Polyhedral Ground Set

In this paper, we study the set $\mathcal{S}^κ= \{ (x,y)\in\mathcal{G}\times\mathbb{R}^n : y_j = x_j^κ, j=1,\dots,n\}$, where $κ> 1$ and the ground set $\mathcal{G}$ is a nonempty polytope contained in $[0,1]^n$. This nonconvex set is closely related to separable standard quadratic programming and appears as a substructure in potential-based network flow problems from gas and water networks. Our aim is to obtain the convex hull of $\mathcal{S}^κ$ or its tight outer-approximation for the special case when the ground set $\mathcal{G}$ is the standard simplex. We propose power cone, second-order cone and semidefinite programming relaxations for this purpose, which are further strengthened by the Reformulation-Linearization Technique and the Reformulation-Perspectification Technique. For $κ=2$, we obtain the convex hull of $\mathcal{S}^κ$ in the low-dimensional setting. For general $κ$, we give approximation guarantees for the power cone representable relaxation, the weakest relaxation we consider. We prove that this weakest relaxation is tight with probability one as $n\to\infty$ when a uniformly generated linear objective is optimized over it. Finally, we provide the results of our extensive computational experiments comparing the empirical strength of several conic programming relaxations that we propose.

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arXiv2025

Geoffrion's theorem beyond finiteness and rationality

Geoffrion's theorem is a fundamental result from mathematical programming assessing the quality of Lagrangian relaxation, a standard technique to get bounds for integer programs. An often implicit condition is that the set of feasible solutions is finite or described by rational linear constraints. However, we show through concrete examples that the conclusion of Geoffrion's theorem does not necessarily hold when this condition is dropped. We then provide sufficient conditions ensuring the validity of the result even when the feasible set is not finite and cannot be described using finitely-many linear constraints.

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arXiv2025

Improving Full Strong Branching Decisions by Incorporating Additional Information

The full strong branching (FSB) rule is well known to produce extremely small branch-and-bound trees. This rule guides branching decisions based exclusively on the information regarding local gains in the linear programming (LP) bounds. We identify and correct two key shortcomings in FSB. First, the LP gains may be overestimations of the improvement in global dual bounds whenever pruning is possible. We propose a modification to address this issue, that incorporates primal bounds and readjusts the relative importance of the larger and smaller LP gains. Second, FSB decisions may be myopic as they consider only local LP gains and cannot foresee the impact of branching decisions on feasibility or integrality beyond immediate children. To address this weakness, we present an approach that detects global asymmetry trends in infeasibility and integrality due to 0 and 1 assignments and incorporates them into the FSB score function. We further extend this approach to achieve more balanced trees even when the branch-and-bound tree prunes primarily by bounds. Using randomly generated problem instances with known structures, we derive insights and fine-tune our modified scores. Evaluation on MIPLIB 2017 Benchmark instances shows a 22-35\% reduction in mean tree sizes for solved cases and a 3.6-5.6\% decrease in the remaining gap for unsolved ones. Our approach extends to reliability branching (RB), where improved scores reduce mean tree sizes by 5-13\% on solved instances and lower the mean gap by 2.6-4.3\% on unsolved instances, depending on primal bound quality.

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arXiv2025

Sensitivity analysis for mixed binary quadratic programming

We consider sensitivity analysis for Mixed Binary Quadratic Programs (MBQPs) with respect to changing right-hand-sides (rhs). We show that even if the optimal solution of a given MBQP is known, it is NP-hard to approximate the change in objective function value with respect to changes in rhs. Next, we study algorithmic approaches to obtaining dual bounds for MBQP with changing rhs. We leverage Burer's completely-positive (CPP) reformulation of MBQPs. Its dual is an instance of co-positive programming (COP), and can be used to obtain sensitivity bounds. We prove that strong duality between the CPP and COP problems holds if the feasible region is bounded or if the objective function is convex, while the duality gap can be strictly positive if neither condition is met. We also show that the COP dual has multiple optimal solutions, and the choice of the dual solution affects the quality of the bounds with rhs changes. We finally provide a method for finding good nearly optimal dual solutions, and we present preliminary computational results on sensitivity analysis for MBQPs.

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arXiv2025

Regularized MIP Model for Integrating Energy Storage Systems and its Application for Solving a Trilevel Interdiction Problem

Incorporating energy storage systems (ESS) into power systems has been studied in many recent works, where binary variables are often introduced to model the complementary nature of battery charging and discharging. A conventional approach for these ESS 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 ESS optimal power flow (OPF) problem. 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 ESS OPF problem, thereby providing a valid and tight upper bound for the ESS OPF problem. The use of the regularized MIP model allows us to solve a trilevel min-max-min network contingency problem which is otherwise intractable to solve.

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arXiv2024

Variable Selection for Kernel Two-Sample Tests

We consider the variable selection problem for two-sample tests, aiming to select the most informative variables to determine whether two collections of samples follow the same distribution. To address this, we propose a novel framework based on the kernel maximum mean discrepancy (MMD). Our approach seeks a subset of variables with a pre-specified size that maximizes the variance-regularized kernel MMD statistic. We focus on three commonly used types of kernels: linear, quadratic, and Gaussian. From a computational perspective, we derive mixed-integer programming formulations and propose exact and approximation algorithms with performance guarantees to solve these formulations. From a statistical viewpoint, we derive the rate of testing power of our framework under appropriate conditions. These results show that the sample size requirements for the three kernels depend crucially on the number of selected variables, rather than the data dimension. Experimental results on synthetic and real datasets demonstrate the superior performance of our method, compared to other variable selection frameworks, particularly in high-dimensional settings.

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arXiv2024

Approximating the Gomory Mixed-Integer Cut Closure Using Historical Data

Many operations related optimization problems involve repeatedly solving similar mixed integer linear programming (MILP) instances with the same constraint matrix but differing objective coefficients and right-hand-side values. The goal of this paper is to generate good cutting-planes for such instances using historical data. Gomory mixed integer cuts (GMIC) for a general MILP can be parameterized by a vector of weights to aggregate the constraints into a single equality constraint, where each such equality constraint in turn yields a unique GMIC. In this paper, we prove that for a family of MILP instances, where the right-hand-side of the instances belongs to a lattice, the GMIC closure for every instance in this infinite family can be obtained using the same finite list of aggregation weights. This result motivates us to build a simple heuristic to efficiently select aggregations for generating GMICs from historical data of similar instances with varying right-hand-sides and objective function coefficients. For testing our method, we generated families of instances by perturbing the right-hand-side and objective functions of MIPLIB 2017 instances. The proposed heuristic can significantly accelerate the performance of Gurobi for many benchmark instances, even when taking into account the time required to predict aggregation multipliers and compute the cut coefficients. To the best of our knowledge, this is the first work in the literature of data-driven cutting plane generation that is able to significantly accelerate the performance of a commercial state-of-the-art MILP solver, using default solver settings, on large-scale benchmark instances.

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arXiv2024

Lagrangian dual with zero duality gap that admits decomposition

For mixed integer programs (MIPs) with block structures and coupling constraints, on dualizing the coupling constraints the resulting Lagrangian relaxation becomes decomposable into blocks which allows for the use of parallel computing. However, the resulting Lagrangian dual can have non-zero duality gap due to the inherent non-convexity of MIPs. In this paper, we propose two reformulations of such MIPs by adding redundant constraints, such that the Lagrangian dual obtained by dualizing the coupling constraints and the redundant constraints have zero duality gap while still remaining decomposable. One of these reformulations is similar, although not the same as the RLT hierarchy. In this case, we present multiplicative bounds on the quality of the dual bound at each level of the hierarchy for packing and covering MIPs. We show our results are applicable to general sparse MIPs, where decomposability is revealed via the tree-decomposition of the intersection graph of the constraint matrix. In preliminary experiments, we observe that the proposed Lagrangian duals give better dual bounds than classical Lagrangian dual and Gurobi in equal time, where Gurobi is not exploiting decomposability.

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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 $k \leq 4$.

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