arXiv subjects

Aida Khajavirad

Publications and source records attributed to Aida Khajavirad.

18 recordsLinked to original sources

arXiv2026

A polynomial-time solvable class of sparse box-constrained polynomial optimization problems

We study the problem of minimizing a multivariate polynomial function over the unit hypercube. Exploiting sparsity in the interaction graph or hypergraph, we identify variables that can be restricted to binary values at optimality and eliminate the remaining continuous variables component-wise, reducing the problem to structured binary polynomial optimization. For quadratic objectives, we obtain exact polynomial-time solvability in the standard Turing bit model under conditions involving treewidth and the nonconvexity of the continuous components. For higher-degree objectives, we obtain an exact polynomial-time algorithm in the unit-cost algebraic RAM model under logarithmic incidence treewidth and small, weakly coupled continuous components.

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arXiv2026

A scalable linear programming-based framework for data clustering

We extend the linear programming-based algorithm of De Rosa et al~\cite{derKhaWan24} for K-means clustering to two important clustering paradigms: fair K-means clustering and spectral clustering. For fair K-means clustering, we show that widely used notions of group fairness can be incorporated into the partition-matrix formulation of K-means clustering through a linear number of linear inequalities. For spectral clustering, we consider a linear programming relaxation of the minimum ratio-cut problem that fits naturally within the same framework. We complement these formulations with problem-specific initialization and rounding procedures and evaluate the resulting algorithms on a large collection of real-world data sets. Denoting by $n$ the number of data points, our computational results demonstrate that the proposed approach solves $90\%$ of benchmark instances with $n \leq 3000$ to within $1\%$ optimality in at most three hours. This in turn demonstrates the remarkable strength of the proposed LP relaxations in both applications. Moreover, for more than $56\%$ of the instances, the proposed algorithm finds better solutions than those produced by popular fair Lloyd-type and spectral clustering heuristics.

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arXiv2026

The complete edge relaxation for binary polynomial optimization

We consider the multilinear polytope, defined as the convex hull of the feasible region of a lifted binary polynomial optimization problem. We define a relaxation in an extended space for this polytope, which we call the complete edge relaxation. The complete edge relaxation is stronger than several well-known relaxations of the multilinear polytope, including the standard linearization, the flower relaxation, and the intersection of all possible recursive McCormick relaxations. In addition, for fixed-degree binary polynomial optimization problems, the case of primary practical interest, the complete edge relaxation is of polynomial size and is computationally efficient in practice. We prove that the complete edge relaxation is an extension of the multilinear polytope if and only if the corresponding hypergraph is alpha-acyclic, the most general type of hypergraph acyclicity. This is in stark contrast with the widely-used standard linearization, which describes the multilinear polytope if and only if the hypergraph is Berge-acyclic, the most restrictive type of hypergraph acyclicity. Finally, we introduce a new class of facet-defining inequalities for the multilinear polytope of alpha-cycles of length three, which serve as the generalization of the well-known triangle inequalities for the Boolean quadric polytope.

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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

On the power of linear programming for K-means clustering

In [SIAM J. Optim., 2022], the authors introduced a new linear programming (LP) relaxation for K-means clustering. In this paper, we further investigate both theoretical and computational properties of this relaxation. As evident from our numerical experiments with both synthetic real-world data sets, the proposed LP relaxation is almost always tight; i.e. its optimal solution is feasible for the original nonconvex problem. To better understand this unexpected behaviour, on the theoretical side, we focus on K-means clustering with two clusters, and we obtain sufficient conditions under which the LP relaxation is tight. We further analyze the sufficient conditions when the input is generated according to a popular stochastic model and obtain recovery guarantees for the LP relaxation. We conclude our theoretical study by constructing a family of inputs for which the LP relaxation is never tight. Denoting by $n$ the number of data points to be clustered, the LP relaxation contains $Ω(n^3)$ inequalities making it impractical for large data sets. To address the scalability issue, by building upon a cutting-plane algorithm together with the GPU implementation of PDLP, a first-order method LP solver, we develop an efficient algorithm that solves the proposed LP and hence the K-means clustering problem, for up to $n \leq 4000$ data points.

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arXiv2026

Tight semidefinite programming relaxations for sparse box-constrained quadratic programs

We introduce a new class of semidefinite programming (SDP) relaxations for sparse box-constrained quadratic programs, obtained by a novel integration of the Reformulation Linearization Technique into standard SDP relaxations while explicitly exploiting the sparsity of the problem. The resulting relaxations are not implied by the existing LP and SDP relaxations for this class of optimization problems. We establish a sufficient condition under which the convex hull of the feasible region of the lifted quadratic program is SDP-representable; the proof is constructive and yields an explicit extended formulation. Although the resulting SDP may be of exponential size in general, we further identify additional structural conditions on the sparsity of the optimization problem that guarantee the existence of a polynomial-size SDP-representable formulation, which can be constructed in polynomial time.

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arXiv2025

Beyond hypergraph acyclicity: limits of tractability for pseudo-Boolean optimization

In this paper, we study the problem of minimizing a polynomial function with literals over all binary points, often referred to as pseudo-Boolean optimization. We investigate the fundamental limits of computation for this problem by providing new necessary conditions and sufficient conditions for tractability. On the one hand, we obtain the first intractability results, in the best-case sense, for pseudo-Boolean optimization problems on signed hypergraphs with bounded rank, in terms of the treewidth of the intersection graph. Namely, first, under some mild assumptions, we show that for every sequence of hypergraphs indexed by the treewidth and with bounded rank, the complexity of solving the associated pseudo-Boolean optimization problem grows super-polynomially in the treewidth. Second, we show that any hypergraph of bounded rank is the underlying hypergraph of some signed hypergraph for which the corresponding pseudo-Boolean polytope has an exponential extension complexity in the treewidth. On the other hand, we introduce the nest-set gap, a new hypergraph-theoretic notion that enables us to define a notion of "distance" from the hypergaph acyclicity. We prove that if this distance is bounded, the pseudo-Boolean polytope admits a polynomial-size extended formulation. This in turn enables us to obtain a polynomial-time algorithm for a large class of pseudo-Boolean optimization problems whose underlying hypergraphs contain beta-cycles.

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arXiv2025

Extended formulations for the multilinear polytope of acyclic hypergraphs

This article provides an overview of our joint work on binary polynomial optimization over the past decade. We define the multilinear polytope as the convex hull of the feasible region of a linearized binary polynomial optimization problem. By representing the multilinear polytope with hypergraphs, we investigate the connections between hypergraph acyclicity and the complexity of the facial structure of the multilinear polytope. We characterize the acyclic hypergraphs for which a polynomial-size extended formulation for the multilinear polytope can be constructed in polynomial time.

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arXiv2024

Inference in higher-order undirected graphical models and binary polynomial optimization

We consider the problem of inference in higher-order undirected graphical models with binary labels. We formulate this problem as a binary polynomial optimization problem and propose several linear programming relaxations for it. We compare the strength of the proposed linear programming relaxations theoretically. Finally, we demonstrate the effectiveness of these relaxations by performing a computational study for two important applications, namely, image restoration and decoding error-correcting codes.

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arXiv2024

The pseudo-Boolean polytope and polynomial-size extended formulations for binary polynomial optimization

With the goal of obtaining strong relaxations for binary polynomial optimization problems, we introduce the pseudo-Boolean polytope defined as the convex hull of the set of binary points satisfying a collection of equations containing pseudo-Boolean functions. By representing the pseudo-Boolean polytope via a signed hypergraph, we obtain sufficient conditions under which this polytope has a polynomial-size extended formulation. Our new framework unifies and extends all prior results on the existence of polynomial-size extended formulations for the convex hull of the feasible region of binary polynomial optimization problems of degree at least three.

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arXiv2024

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 semi-random 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.

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arXiv2024

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

We consider the problem of packing 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 (JOGO, 2009). Our results serve as a theoretical justification for the ineffectiveness of existing machinery for convexification of non-overlapping constraints.

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arXiv2024

The ratio-cut polytope and K-means clustering

We introduce the ratio-cut polytope defined as the convex hull of ratio-cut vectors corresponding to all partitions of $n$ points in $\mathbb R^m$ into at most $K$ clusters. This polytope is closely related to the convex hull of the feasible region of a number of clustering problems such as K-means clustering and spectral clustering. We study the facial structure of the ratio-cut polytope and derive several types of facet-defining inequalities. We then consider the problem of K-means clustering and introduce a novel linear programming (LP) relaxation for it. Subsequently, we focus on the case of two clusters and derive sufficient conditions under which the proposed LP relaxation recovers the underlying clusters exactly. Namely, we consider the stochastic ball model, a popular generative model for K-means clustering, and we show that if the separation distance between cluster centers satisfies $Δ> 1+\sqrt 3$, then the LP relaxation recovers the planted clusters with high probability. This is a major improvement over the only existing recovery guarantee for an LP relaxation of K-means clustering stating that recovery is possible with high probability if and only if $Δ> 4$. Our numerical experiments indicate that the proposed LP relaxation significantly outperforms a popular semidefinite programming relaxation in recovering the planted clusters.

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arXiv2023

A polynomial-size extended formulation for the multilinear polytope of beta-acyclic hypergraphs

We consider the multilinear polytope defined as the convex hull of the set of binary points satisfying a collection of multilinear equations. The complexity of the facial structure of the multilinear polytope is closely related to the acyclicity degree of the underlying hypergraph. We obtain a polynomial-size extended formulation for the multilinear polytope of beta-acyclic hypergraphs, hence characterizing the acyclic hypergraphs for which such a formulation can be constructed.

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arXiv2023

Explicit convex hull description of bivariate quadratic sets with indicator variables

We consider the nonconvex set $\mathcal S_n = \{(x,X,z): X = x x^T, \; x (1-z) =0,\; x \geq 0,\; z \in \{0,1\}^n\}$, which is closely related to the feasible region of several difficult nonconvex optimization problems such as the best subset selection and constrained portfolio optimization. Utilizing ideas from convex analysis and disjunctive programming, we obtain an explicit description for the closure of the convex hull of $\mathcal S_2$ in the space of original variables. In order to generate valid inequalities corresponding to supporting hyperplanes of the convex hull of $\mathcal S_2$, we present a simple separation algorithm that can be incorporated in branch-and-cut based solvers to enhance the quality of existing relaxations.

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arXiv2023

On the strength of recursive McCormick relaxations for binary polynomial optimization

Recursive McCormick relaxations have been among the most popular convexification techniques for binary polynomial optimization problems. It is well-understood that both the quality and the size of these relaxations depend on the recursive sequence, and finding an optimal recursive sequence amounts to solving a difficult combinatorial optimization problem. In this paper, we prove that any recursive McCormick relaxation is implied by the extended flower relaxation, a linear programming relaxation that is a natural generalization of the flower relaxation introduced by Del Pia and Khajavirad 2018, which for binary polynomial optimization problems with fixed degree can be solved in strongly polynomial time.

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arXiv2022

Efficient Joint Object Matching via Linear Programming

Joint object matching, also known as multi-image matching, namely, the problem of finding consistent partial maps among all pairs of objects within a collection, is a crucial task in many areas of computer vision. This problem subsumes bipartite graph matching and graph partitioning as special cases and is NP-hard, in general. We develop scalable linear programming (LP) relaxations with theoretical performance guarantees for joint object matching. We start by proposing a new characterization of consistent partial maps; this in turn enables us to formulate joint object matching as an integer linear programming (ILP) problem. To construct strong LP relaxations, we study the facial structure of the convex hull of the feasible region of this ILP, which we refer to as the joint matching polytope. We present an exponential family of facet-defining inequalities that can be separated in strongly polynomial time, hence obtaining a partial characterization of the joint matching polytope that is both tight and cheap to compute. To analyze the theoretical performance of the proposed LP relaxations, we focus on permutation group synchronization, an important special case of joint object matching. We show that under the random corruption model for the input maps, a simple LP relaxation, that is, an LP containing only a very small fraction of the proposed facet-defining inequalities, recovers the ground truth with high probability if the corruption level is below $40\%$. Finally, via a preliminary computational study on synthetic data, we show that the proposed LP relaxations outperform a popular SDP relaxation both in terms of recovery and tightness.

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arXiv2022

Linear Programming and Community Detection

The problem of community detection with two equal-sized communities is closely related to the minimum graph bisection problem over certain random graph models. In the stochastic block model distribution over networks with community structure, a well-known semidefinite programming (SDP) relaxation of the minimum bisection problem recovers the underlying communities whenever possible. Motivated by their superior scalability, we study the theoretical performance of linear programming (LP) relaxations of the minimum bisection problem for the same random models. We show that unlike the SDP relaxation that undergoes a phase transition in the logarithmic average-degree regime, the LP relaxation exhibits a transition from recovery to non-recovery in the linear average-degree regime. We show that in the logarithmic average-degree regime, the LP relaxation fails in recovering the planted bisection with high probability.

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