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This paper examines nonlinear optimization problems that incorporate discrete decisions. We introduce new improved formulation techniques that take advantage of the simplotope structure present in the domain of the binarization variables. Our technique identifies new polynomially solvable instances for price promotion problem initially studied by Cohen et al. (2021) and allows us to develop a linear programming (LP) formulation for inventory optimization problem under a choice model proposed by Boada-Collado and Martinez-de Albeniz (2020). The techniques we develop rely on ideal formulations for submodular and fractional compositions of discrete functions, improving prior formulations for bilinear products suggested by Adams and Henry (2012). Submodular compositions also generalize L natural functions over bounded domains and our construction provides new insights into Lovasz-extension based formulations for this class of problems while expanding the class of nonlinear discrete optimization problems amenable to linear programming based techniques.

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.

We introduce new relaxation hierarchies for facial disjunctive programs (FD) and concavo-convex programs, where the latter class of problems includes disjoint bilinear programming (DBP) and concave minimization (CM) as special cases. Meanwhile, FD restricts that feasible solutions belong to a collection of faces of a Cartesian product of polytopes and generalizes 0-1 programs. We construct these relaxation hierarchies by utilizing rational functions that are barycentric coordinates for polytopes and derive these expressions using the double-description (DD) procedure. The hierarchies, which have geometric and algebraic underpinnings, converge to the convex hull for these problems at a finite level. Our hierarchy provides the first unifying framework to analyze and tighten relaxations from disjunctive programming (DP) and reformulation-linearization technique (RLT) for these problem classes.

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.

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., Comment: arXiv admin note: text overlap with arXiv:2205.01442

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.

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.

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., Comment: 37 pages, 12 figures

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., Comment: 50 pages, 12 figures

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 $50\times 50$., Comment: 3 Figures

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.