arXiv subjects

Mohit Tawarmalani

Publications and source records attributed to Mohit Tawarmalani.

15 recordsLinked to original sources

arXiv2026

Discreteness to Convexity: Promotion Planning via Simplotope Triangulation

Price promotion optimization is a computationally challenging problem central to supermarket operations, requiring simultaneous pricing decisions across multiple products and periods. This paper introduces a new formulation for price promotion by developing convex hull results for supermodular compositions of univariate functions over a simplotope. Leveraging this reformulation with Gurobi, we achieve substantial performance gains: instances with up to 125 products, 20 periods, and 5 price levels are solved in an average of 7 minutes, demonstrating the potential to handle even larger instances. Our exact solution methods extract 25--48\% additional profit from promotion planning relative to state-of-the-art heuristic approaches. Additionally, we extend the polynomially solvable cases from two to multiple price levels and expand our results to allow for multiplicative historical effects. Our core methodological innovation applies to a broad class of nonlinear discrete optimization problems. Specifically, our results convexify a class of nonlinear functions that includes monomials and the widely studied L natural function structure.

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arXiv2026

Disjunctive Submodular Functions: Envelopes and Applications to Inventory and 0-1 Quadratic Optimization

This paper considers convex envelopes of disjunctive submodular functions---functions that are lattice family submodular over faces of a hypercube---and constructs the first strongly polynomial algorithm for their separation when there are two facial disjunctions. Submodular functions, whose convex envelopes are characterized by the Lovász extension, have occupied a fundamental role in constructing relaxations for combinatorial and nonlinear optimization problems. However, disjunctive submodular function envelopes have not been explored besides the use of ellipsoid algorithm, which remains practically intractable. Our algorithm is derived in three steps by expressing the disjunctive function as a minimum of two extended submodular functions, introducing a variable lifting technique, and constructing the sublinear envelope in the lifted space. The paper also makes several other contributions. First, we provide a disjunctive formulation for the case where each submodular function admits a linear programming formulation. Second, we derive the closed-form sublinear envelope characterization for intersecting submodular functions, yielding new structural insights into a multi-product inventory sales maximization problem. Third, we fully characterize the convex envelope of a bilinear function defined over a cycle graph in the original variable space. Finally, we show computationally that the cycle inequalities close approximately 60\% of the gap for complete and Hadamard graphs, over 30\% of the gap for complete bipartite graphs, and over 80\% of the gap for sparse graphs such as cactus and Halin graphs. The resulting relaxations are also more efficient to solve than previous extended space formulations.

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arXiv2026

Axis-Aligned Relaxations for Mixed-Integer Nonlinear Programming

We present a novel relaxation framework for general mixed-integer nonlinear programming (MINLP) grounded in computational geometry. Our approach constructs polyhedral relaxations by convexifying finite sets of strategically chosen points, iteratively refining the approximation to converge toward the simultaneous convex hull of factorable function graphs. The framework is underpinned by three key contributions: (i) a new class of explicit inequalities for products of functions that strictly improve upon standard factorable and composite relaxation schemes; (ii) a proof establishing that the simultaneous convex hull of multilinear functions over axis-aligned regions is fully determined by their values at corner points, thereby generalizing existing results from hypercubes to arbitrary axis-aligned domains; and (iii) the integration of computational geometry tools, specifically voxelization and QuickHull, to efficiently approximate feasible regions and function graphs. We implement this framework and evaluate it on randomly generated polynomial optimization problems and a suite of 619 instances from \texttt{MINLPLib}. Numerical results demonstrate significant improvements over state-of-the-art benchmarks: on polynomial instances, our relaxation closes an additional 20--25\% of the optimality gap relative to standard methods on half the instances. Furthermore, compared against an enhanced factorable programming baseline and Gurobi's root-node bounds, our approach yields superior dual bounds on approximately 30\% of \texttt{MINLPLib} instances, with roughly 10\% of cases exhibiting a gap reduction exceeding 50\%.

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arXiv2025

Minimizing risk measures with applications in network traffic engineering

This paper presents a novel two-stage optimization framework designed to model integrated quantile functions, which leads to the formulation of a bilinear optimization problem (P). A specific instance of this framework offers a new approach to minimizing the Value-at-risk (Var) and the Conditional Value-at-risk (CVar), thus providing a broader perspective on risk assessment and optimization. We investigate various convexification techniques to under- and over-estimate the optimal value of (P), resulting in new and tighter lower- and upper-convex estimators for the Var minimization problems. Furthermore, we explore the properties and implications of the bilinear optimization problem (P) in connection to the integrated quantile functions. Finally, to illustrate the practical applications of our approach, we present computational comparisons in the context of real-life network traffic engineering problems, demonstrating the effectiveness of our proposed framework.

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arXiv2025

New finite relaxation hierarchies for concavo-convex, disjoint bilinear programs, and facial disjunctions

This paper introduces novel relaxation hierarchies for concavo-convex programs (CXP), a class of problems that includes disjoint bilinear programming (DBP) and concave minimization (CM) as special cases. At the core of these hierarchies is an algorithm based on double-description (DD) that computes the barycentric coordinates of a polyhedral cone as rational, non-negative functions representing multipliers associated with the cone's rays. These hierarchies combine geometric structure derived from barycentric coordinates with algebraic techniques via rational functions, achieving the convex hull in $m$ iterations, where $m$ is the number of inequalities that a subset of the variables must satisfy. Our framework offers the first unified approach to analyze and tighten relaxations from disjunctive programming (DP) and reformulation-linearization technique (RLT) for CXP. We also demonstrate that our methods extend to facial disjunctive programs (FDP), where solutions are constrained to lie on faces of a Cartesian product of polytopes, generalizing known hierarchies for 0-1 programs.

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arXiv2025

Minimum reflux calculation for multicomponent distillation in multi-feed, multi-product columns: Algorithms and examples

In this work, we present the first algorithm for identifying the minimum reboiler vapor duty requirement for a general multi-feed, multi-product (MFMP) distillation column separating ideal multicomponent mixtures. This algorithm incorporates our latest advancement in developing the first shortcut model for MFMP columns. We demonstrate the accuracy and efficiency of this algorithm through case studies. The results obtained from these case studies also provide valuable insights on optimal design of MFMP columns. Many of these insights are against the existing design guidelines and heuristics. For example, placing a colder saturated feed stream above a hotter saturated feed stream sometimes leads to higher energy requirement. Furthermore, decomposing a general MFMP column into individual simple columns may lead to incorrect estimation of the minimum reflux ratio for the MFMP column. Thus, the algorithm presented here offers a fast, accurate, and automated approach to synthesize new, energy-efficient, and cost-effective MFMP columns.

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arXiv2024

Active Learning for Fair and Stable Online Allocations

We explore an active learning approach for dynamic fair resource allocation problems. Unlike previous work that assumes full feedback from all agents on their allocations, we consider feedback from a select subset of agents at each epoch of the online resource allocation process. Despite this restriction, our proposed algorithms provide regret bounds that are sub-linear in number of time-periods for various measures that include fairness metrics commonly used in resource allocation problems and stability considerations in matching mechanisms. The key insight of our algorithms lies in adaptively identifying the most informative feedback using dueling upper and lower confidence bounds. With this strategy, we show that efficient decision-making does not require extensive feedback and produces efficient outcomes for a variety of problem classes.

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arXiv2024

MIP Relaxations in Factorable Programming

In this paper, we develop new discrete relaxations for nonlinear expressions in factorable programming. We utilize specialized convexification results as well as composite relaxations to develop mixed-integer programming (MIP) relaxations. Our relaxations rely on ideal formulations of convex hulls of outer-functions over a combinatorial structure that captures local inner-function structure. The resulting relaxations often require fewer variables and are tighter than currently prevalent ones. Finally, we provide computational evidence to demonstrate that our relaxations close approximately 60-70% of the gap relative to McCormick relaxations and significantly improves the relaxations used in a state-of-the-art solver on various instances involving polynomial functions.

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arXiv2024

Convexification Techniques for Fractional Programs

This paper develops a correspondence relating convex hulls of fractional functions with those of polynomial functions over the same domain. Using this result, we develop a number of new reformulations and relaxations for fractional programming problems. First, we relate 0-1 problems involving a ratio of affine functions with the boolean quadric polytope, and use inequalities for the latter to develop tighter formulations for the former. Second, we derive a new formulation to optimize a ratio of quadratic functions over a polytope using copositive programming. Third, we show that univariate fractional functions can be convexified using moment hulls. Fourth, we develop a new hierarchy of relaxations that converges finitely to the simultaneous convex hull of a collection of ratios of affine functions of 0-1 variables. Finally, we demonstrate theoretically and computationally that our techniques close a significant gap relative to state-of-the-art relaxations, require much less computational effort, and can solve larger problem instances.

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arXiv2022

Extracting structure from functional expressions for continuous and discrete relaxations of MINLP

In this paper, we develop new continuous and discrete relaxations for nonlinear expressions in an MINLP. In contrast to factorable programming, our techniques utilize the inner-function structure by encapsulating it in a polyhedral set, using a technique first proposed in [12]. We tighten the relaxations derived in [33,13] and obtain new relaxations for functions that could not be treated using prior techniques. We develop new discretization-based mixed-integer programming relaxations that yield tighter relaxations than similar relaxations in the literature. These relaxations utilize the simplotope that captures inner-function structure to generalize the incremental formulation of [8] to multivariate functions. In particular, when the outer-function is supermodular, our formulations require exponentially fewer continuous variables than any previously known formulation.

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arXiv2021

Convexification of Permutation-Invariant Sets and an Application to Sparse PCA

We develop techniques to convexify a set that is invariant under permutation and/or change of sign of variables and discuss applications of these results. First, we convexify the intersection of the unit ball of a permutation and sign-invariant norm with a cardinality constraint. This gives a nonlinear formulation for the feasible set of sparse principal component analysis (sparse PCA) and an alternative proof of the $K$-support norm. Second, we characterize the convex hull of sets of matrices defined by constraining their singular values. As a consequence, we generalize an earlier result that characterizes the convex hull of rank-constrained matrices whose spectral norm is below a given threshold. Third, we derive convex and concave envelopes of various permutation-invariant nonlinear functions and their level-sets over hypercubes, with congruent bounds on all variables. Finally, we develop new relaxations for the exterior product of sparse vectors. Using these relaxations for sparse PCA, we show that our relaxation closes $98\%$ of the gap left by a classical SDP relaxation for instances where the covariance matrices are of dimension up to $50\times 50$.

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arXiv2021

FloMore: Meeting bandwidth requirements of flows

Wide-area cloud provider networks must support the bandwidth requirements of diverse services (e.g., applications, product groups, customers) despite failures. Existing traffic engineering (TE) schemes operate at much coarser granularity than services, which we show necessitates unduly conservative decisions. To tackle this, we present FloMore, which directly considers the bandwidth needs of individual services and ensures they are met a desired percentage of time. Rather than meet the requirements for all services over the same set of failure states, FloMore exploits a key opportunity that each service could meet its bandwidth requirements over a different set of failure states. FloMore consists of an offline phase that identifies the critical failure states of each service, and on failure allocates traffic in a manner that prioritizes those services for which that failure state is critical. We present a novel decomposition scheme to handle FloMore's offline phase in a tractable manner. Our evaluations show that FloMore outperforms state-of-the-art TE schemes including SMORE and Teavar, and also out-performs extensions of these schemes that we devise. The results also show FloMore's decomposition approach allows it to scale well to larger network topologies.

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arXiv2021

Optimal Design of Membrane Cascades for Gaseous and Liquid Mixtures via MINLP

Given the growing concern of reducing CO2 emissions, it is desirable to identify, for a given separation carried out through a membrane cascade, the optimum design that yields the lowest energy consumption. Nevertheless, designing a membrane cascade is challenging since, there are often multiple feasible configurations that differ in their energy consumption and cost. In this work, we develop a Mixed Integer Non-linear Program (MINLP) that, for a given binary separation, which may be either liquid or gaseous, finds the cascade and its operating conditions that minimize energy consumption. To model the separation at each membrane in the cascade, we utilize the analytical solution of a system of differential and algebraic equations derived from the crossflow model and the solution-diffusion theory. We provide numerical evidence which shows that our single-stage membrane model accurately predicts experimental data. Unfortunately, the resulting membrane model is non-convex and, even state-of-the-art solvers struggle to prove global optimality of the cascades and the operating conditions identified. In this paper, we derive various cuts that help with relaxation quality and, consequently, accelerate convergence of branch-and-bound based solvers. More specifically, we demonstrate, on various examples, that our cuts help branch-and-bound solvers converge within 5\% optimality gap in a reasonable amount of time and such a tolerance level was not achieved by a simple formulation of the membrane model. The proposed optimization model is an easy-to-use tool for practitioners and researchers to design energy efficient membrane cascades.

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arXiv2020

Advances in MINLP to Identify Energy-efficient Distillation Configurations

In this paper, we describe the first mixed-integer nonlinear programming (MINLP) based solution approach that successfully identifies the most energy-efficient distillation configuration sequence for a given separation. Current sequence design strategies are largely heuristic. The rigorous approach presented here can help reduce the significant energy consumption and consequent greenhouse gas emissions by separation processes, where crude distillation alone is estimated to consume 6.9 quads of energy per year globally. The challenge in solving this problem arises from the large number of feasible configuration sequences and because the governing equations contain non-convex fractional terms. We make several advances to enable solution of these problems. First, we model discrete choices using a formulation that is provably tighter than previous formulations. Second, we highlight the use of partial fraction decomposition alongside Reformulation-Linearization Technique (RLT). Third, we obtain convex hull results for various special structures. Fourth, we develop new ways to discretize the MINLP. Finally, we provide computational evidence to demonstrate that our approach significantly outperforms the state-of-the-art techniques.

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arXiv2017

Information Theoretic Limits for Linear Prediction with Graph-Structured Sparsity

We analyze the necessary number of samples for sparse vector recovery in a noisy linear prediction setup. This model includes problems such as linear regression and classification. We focus on structured graph models. In particular, we prove that sufficient number of samples for the weighted graph model proposed by Hegde and others is also necessary. We use the Fano's inequality on well constructed ensembles as our main tool in establishing information theoretic lower bounds.

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