In the present work, we consider Zuckerberg’s method for geometric convex-hull proofs introduced in Zuckerberg (Oper Res Lett 44(5):625–629, 2016). It has only been scarcely adopted in the literature so far, despite the great flexibility in designing algorithmic proofs for the completeness of polyhedral descriptions that it offers. We suspect that this is partly due to the rather heavy algebraic framework

We consider so called 2-stage stochastic integer programs (IPs) and their generalized form, so called multi-stage stochastic IPs. A 2-stage stochastic IP is an integer program of the form \(\max \{ c^T x \mid {\mathcal {A}} x = b, \,l \le x \le u,\, x \in {\mathbb {Z}}^{s + nt} \}\) where the constraint matrix \({\mathcal {A}} \in {\mathbb {Z}}^{r n \times s +nt}\) consists roughly of n repetitions

The price of anarchy has become a standard measure of the efficiency of equilibria in games. Most of the literature in this area has focused on establishing worst-case bounds for specific classes of games, such as routing games or more general congestion games. Recently, the price of anarchy in routing games has been studied as a function of the traffic demand, providing asymptotic results in light

We consider a parametric family of quadratically constrained quadratic programs and their associated semidefinite programming (SDP) relaxations. Given a nominal value of the parameter at which the SDP relaxation is exact, we study conditions (and quantitative bounds) under which the relaxation will continue to be exact as the parameter moves in a neighborhood around the nominal value. Our framework

For nonconvex optimization problems, possibly having mixed-integer variables, a convergent primal-dual solution algorithm is proposed. The approach applies a proximal bundle method to certain augmented Lagrangian dual that arises in the context of the so-called generalized augmented Lagrangians. We recast these Lagrangians into the framework of a classical Lagrangian by means of a special reformulation

In cooperative games, players have a possibility to form different coalitions. This leads to the questions about ways to motivate all players to collaborate, i.e. to motivate the players to form the so-called grand coalition. One of such ways is captured by the concept of nucleolus, which dates back to Babylonian Talmud. Weighted voting games form a class of cooperative games, that are often used to

This paper considers the problem of solving a special quartic–quadratic optimization problem with a single sphere constraint, namely, finding a global and local minimizer of \(\frac{1}{2}\mathbf {z}^{*}A\mathbf {z}+\frac{\beta }{2}\sum _{k=1}^{n}|z_{k}|^{4}\) such that \(\Vert \mathbf {z}\Vert _{2}=1\). This problem spans multiple domains including quantum mechanics and chemistry sciences and we investigate

In this paper, using an optimal partition approach, we study the parametric analysis of a second-order conic optimization problem, where the objective function is perturbed along a fixed direction. We characterize the notions of so-called invariancy set and nonlinearity interval, which serve as stability regions of the optimal partition. We then propose, under the strict complementarity condition,

In this paper, we bound the integrality gap and the approximation ratio for maximum plane multiflow problems and deduce bounds on the flow-multicut-gap. We consider instances where the union of the supply and demand graphs is planar and prove that there exists a multiflow of value at least half the capacity of a minimum multicut. We then show how to convert any multiflow into a half-integer flow of

Integer linear programs of configurations, or configuration IPs, are a classical tool in the design of algorithms for scheduling and packing problems where a set of items has to be placed in multiple target locations. Herein, a configuration describes a possible placement on one of the target locations, and the IP is used to choose suitable configurations covering the items. We give an augmented IP

Cut generation and lifting are key components for the performance of state-of-the-art mathematical programming solvers. This work proposes a new general cut-and-lift procedure that exploits the combinatorial structure of 0–1 problems via a binary decision diagram (BDD) encoding of their constraints. We present a general framework that can be applied to a wide range of binary optimization problems and

Many practical integer programming problems involve variables with one or two-sided bounds. Dunkel and Schulz (A refined Gomory–Chvátal closure for polytopes in the unit cube, http://www.optimization-online.org/DB_FILE/2012/03/3404.pdf, 2012) considered a strengthened version of Chvátal–Gomory (CG) inequalities that use 0–1 bounds on variables, and showed that the set of points in a rational polytope

Semidefinite programming (SDP) with diagonal constraints arise in many optimization problems, such as Max-Cut, community detection and group synchronization. Although SDPs can be solved to arbitrary precision in polynomial time, generic convex solvers do not scale well with the dimension of the problem. In order to address this issue, Burer and Monteiro (Math Program 95(2):329–357, 2003) proposed to

We study the nonlinear newsvendor problem concerning goods of a non-discrete nature, and a class of stochastic dynamic programs with several application areas such as supply chain management and economics. The class is characterized by continuous state and action spaces, either convex or monotone cost functions that are accessed via value oracles, and affine transition functions. We establish that

We study a Stackelberg game with multiple leaders and a continuum of followers that are coupled via congestion effects. The followers’ problem constitutes a nonatomic congestion game, where a population of infinitesimal players is given and each player chooses a resource. Each resource has a linear cost function which depends on the congestion of this resource. The leaders of the Stackelberg game each

In this paper, we address the minimum-cost node-capacitated multiflow problem in undirected networks. For this problem, Babenko and Karzanov (JCO 24: 202–228, 2012) showed strong polynomial-time solvability via the ellipsoid method. Our result is the first combinatorial polynomial-time algorithm for this problem. Our algorithm finds a half-integral minimum-cost maximum multiflow in \(O(m \log (nCD)\mathrm

For two integers \(k>0\) and \(\ell \), a graph \(G=(V,E)\) is called \((k,\ell )\)-tight if \(|E|=k|V|-\ell \) and \(i_G(X)\le k|X|-\ell \) for each \(X\subseteq V\) for which \(i_G(X)\ge 1\), where \(i_G(X)\) denotes the number of induced edges by X. G is called \((k,\ell )\)-redundant if \(G-e\) has a spanning \((k,\ell )\)-tight subgraph for all \(e\in E\). We consider the following augmentation

A framework is proposed for solving general convex quadratic programs (CQPs) from an infeasible starting point by invoking an existing feasible-start algorithm tailored for inequality-constrained CQPs. The central tool is an exact penalty function scheme equipped with a penalty-parameter updating rule. The feasible-start algorithm merely has to satisfy certain general requirements, and so is the updating

A particularly important substructure in modeling joint linear chance-constrained programs with random right-hand sides and finite sample space is the intersection of mixing sets with common binary variables (and possibly a knapsack constraint). In this paper, we first revisit basic mixing sets by establishing a strong and previously unrecognized connection to submodularity. In particular, we show

The most important ingredient for solving mixed-integer nonlinear programs (MINLPs) to global \(\epsilon \)-optimality with spatial branch and bound is a tight, computationally tractable relaxation. Due to both theoretical and practical considerations, relaxations of MINLPs are usually required to be convex. Nonetheless, current optimization solvers can often successfully handle a moderate presence

We consider optimization problems with an objective function that is estimable using a Monte Carlo oracle, constraint functions that are known deterministically through a constraint-satisfaction oracle, and integer decision variables. Seeking an appropriately defined local minimum, we propose an iterative adaptive sampling algorithm that, during each iteration, performs a statistical local optimality

A correction to this paper has been published: https://doi.org/10.1007/s10107-021-01684-5

Gradient-based optimization algorithms can be studied from the perspective of limiting ordinary differential equations (ODEs). Motivated by the fact that existing ODEs do not distinguish between two fundamentally different algorithms—Nesterov’s accelerated gradient method for strongly convex functions (NAG-SC) and Polyak’s heavy-ball method—we study an alternative limiting process that yields high-resolution

We consider the global optimization of nonconvex mixed-integer quadratic programs with linear equality constraints. In particular, we present a new class of convex quadratic relaxations which are derived via quadratic cuts. To construct these quadratic cuts, we solve a separation problem involving a linear matrix inequality with a special structure that allows the use of specialized solution algorithms

We consider a general class of binary packing problems with a convex quadratic knapsack constraint. We prove that these problems are \(\mathsf {APX}\)-hard to approximate and present constant-factor approximation algorithms based upon two different algorithmic techniques: a rounding technique tailored to a convex relaxation in conjunction with a non-convex relaxation, and a greedy strategy. We further

This article studies the solutions of time-dependent differential inclusions which is motivated by their utility in optimization algorithms and the modeling of physical systems. The differential inclusion is described by a time-dependent set-valued mapping having the property that, for a given time instant, the set-valued mapping describes a maximal monotone operator. By successive application of a

In the classic maximum coverage problem, we are given subsets \(T_1, \ldots , T_m\) of a universe [n] along with an integer k and the objective is to find a subset \(S \subseteq [m]\) of size k that maximizes \(C(S) :=|\cup _{i \in S} T_i|\). It is well-known that the greedy algorithm for this problem achieves an approximation ratio of \((1-e^{-1})\) and there is a matching inapproximability result

An instance of colorful k-center consists of points in a metric space that are colored red or blue, along with an integer k and a coverage requirement for each color. The goal is to find the smallest radius \(\rho \) such that there exist balls of radius \(\rho \) around k of the points that meet the coverage requirements. The motivation behind this problem is twofold. First, from fairness considerations:

We consider quadratic optimization in variables (x, y) where \(0\le x\le y\), and \(y\in \{ 0,1 \}^n\). Such binary variables are commonly referred to as indicator or switching variables and occur commonly in applications. One approach to such problems is based on representing or approximating the convex hull of the set \(\{ (x,xx^T, yy^T)\,:\,0\le x\le y\in \{ 0,1 \}^n \}\). A representation for the

In this paper, we consider an optimization problem involving crashing an activity network under a single disruption. A disruption is an event whose magnitude and timing are random. When a disruption occurs, the duration of an activity that has yet to start—or alternatively, yet to complete—can change. We formulate a two-stage stochastic mixed-integer program, in which the timing of the stage is random

We consider the capacitated cycle covering problem: given an undirected, complete graph G with metric edge lengths and demands on the vertices, we want to cover the vertices with vertex-disjoint cycles, each serving a demand of at most one. The objective is to minimize a linear combination of the total length and the number of cycles. This problem is closely related to the capacitated vehicle routing

There has been a long history of using ordinary differential equations (ODEs) to understand the dynamics of discrete-time algorithms (DTAs). Surprisingly, there are still two fundamental and unanswered questions: (i) it is unclear how to obtain a suitable ODE from a given DTA, and (ii) it is unclear the connection between the convergence of a DTA and its corresponding ODEs. In this paper, we propose

We address the problem of prescribing an optimal decision in a framework where the cost function depends on uncertain problem parameters that need to be learned from data. Earlier work proposed prescriptive formulations based on supervised machine learning methods. These prescriptive methods can factor in contextual information on a potentially large number of covariates to take context specific actions

A flag is a sequence of nested subspaces. Flags are ubiquitous in numerical analysis, arising in finite elements, multigrid, spectral, and pseudospectral methods for numerical pde; they arise in the form of Krylov subspaces in matrix computations, and as multiresolution analysis in wavelets constructions. They are common in statistics too—principal component, canonical correlation, and correspondence

We study the geometry of convex optimization problems given in a Domain-Driven form and categorize possible statuses of these problems using duality theory. Our duality theory for the Domain-Driven form, which accepts both conic and non-conic constraints, lets us determine and certify statuses of a problem as rigorously as the best approaches for conic formulations (which have been demonstrably very

In this paper, we consider the problem of computing the rank of a block-structured symbolic matrix (a generic partitioned matrix) \(A = (A_{\alpha \beta }x_{\alpha \beta })\), where \(A_{\alpha \beta }\) is a \(2 \times 2\) matrix over a field \(\mathbf {F}\) and \(x_{\alpha \beta }\) is an indeterminate for \(\alpha = 1,2,\dots , \mu \) and \(\beta = 1,2, \dots , \nu \). This problem can be viewed

Wasserstein balls, which contain all probability measures within a pre-specified Wasserstein distance to a reference measure, have recently enjoyed wide popularity in the distributionally robust optimization and machine learning communities to formulate and solve data-driven optimization problems with rigorous statistical guarantees. In this technical note we prove that the Wasserstein ball is weakly

A correction to this paper has been published: https://doi.org/10.1007/s10107-021-01668-5

In this paper, we study the lower iteration complexity bounds for finding the saddle point of a strongly convex and strongly concave saddle point problem: \(\min _x\max _yF(x,y)\). We restrict the classes of algorithms in our investigation to be either pure first-order methods or methods using proximal mappings. For problems with gradient Lipschitz constants (\(L_x, L_y\) and \(L_{xy}\)) and strong

We develop a decomposition algorithm for distributionally-robust two-stage stochastic mixed-integer convex conic programs, and its important special case of distributionally-robust two-stage stochastic mixed-integer second order conic programs. This generalizes the algorithm proposed by Sen and Sherali [Mathematical Programming 106(2): 203–223, 2006]. We show that the proposed algorithm is finitely

Many iterative methods in applied mathematics can be thought of as fixed-point iterations, and such algorithms are usually analyzed analytically, with inequalities. In this paper, we present a geometric approach to analyzing contractive and nonexpansive fixed point iterations with a new tool called the scaled relative graph. The SRG provides a correspondence between nonlinear operators and subsets

Let G be a digraph where every node has preferences over its incoming edges. The preferences of a node extend naturally to preferences over branchings, i.e., directed forests; a branching B is popular if B does not lose a head-to-head election (where nodes cast votes) against any branching. Such popular branchings have a natural application in liquid democracy. The popular branching problem is to decide

We introduce an iterative framework for solving graph coloring problems using decision diagrams. The decision diagram compactly represents all possible color classes, some of which may contain edge conflicts. In each iteration, we use a constrained minimum network flow model to compute a lower bound and identify conflicts. Infeasible color classes associated with these conflicts are removed by refining

We study robustness properties of some iterative gradient-based methods for strongly convex functions, as well as for the larger class of functions with sector-bounded gradients, under a relative error model. Proofs of the corresponding convergence rates are based on frequency-domain criteria for the stability of nonlinear systems. Applications are given to inexact versions of gradient descent and

We introduce and study the problem Flexible Graph Connectivity, which in contrast to many classical connectivity problems features a non-uniform failure model. We distinguish between safe and unsafe resources and postulate that failures can only occur among the unsafe resources. Given an undirected edge-weighted graph and a set of unsafe edges, the task is to find a minimum-cost subgraph that remains

A classic result by Cook, Gerards, Schrijver, and Tardos provides an upper bound of \(n \Delta \) on the proximity of optimal solutions of an Integer Linear Programming problem and its standard linear relaxation. In this bound, n is the number of variables and \(\Delta \) denotes the maximum of the absolute values of the subdeterminants of the constraint matrix. Hochbaum and Shanthikumar, and Werman

Lattice-free gradient polyhedra can be used to certify optimality for mixed integer convex minimization models. We consider how to construct these polyhedra for unconstrained models with two integer variables under the assumption that all level sets are bounded. In this setting, a classic result of Bell, Doignon, and Scarf states the existence of a lattice-free gradient polyhedron with at most four

We consider mixed-integer optimal control problems with combinatorial constraints that couple over time such as minimum dwell times. We analyze a lifting and decomposition approach into a mixed-integer optimal control problem without combinatorial constraints and a mixed-integer problem for the combinatorial constraints in the control space. Both problems can be solved very efficiently with existing

The weighted \({\mathcal {T}}\)-free 2-matching problem is the following problem: given an undirected graph G, a weight function on its edge set, and a set \({\mathcal {T}}\) of triangles in G, find a maximum weight 2-matching containing no triangle in \({\mathcal {T}}\). When \({\mathcal {T}}\) is the set of all triangles in G, this problem is known as the weighted triangle-free 2-matching problem

A correction to this paper has been published: https://doi.org/10.1007/s10107-021-01648-9

We study the sparsity of the solutions to systems of linear Diophantine equations with and without non-negativity constraints. The sparsity of a solution vector is the number of its nonzero entries, which is referred to as the \(\ell _0\)-norm of the vector. Our main results are new improved bounds on the minimal \(\ell _0\)-norm of solutions to systems \(A\varvec{x}=\varvec{b}\), where \(A\in \mathbb

In this paper we reconsider a known technique for constructing strong MIP formulations for disjunctive constraints of the form \(x \in \bigcup _{i=1}^m P_i\), where the \(P_i\) are polytopes. The formulation is based on the Cayley Embedding of the union of polytopes, namely, \(Q := \mathrm {conv}(\bigcup _{i=1}^m P_i\times \{\epsilon ^i\})\), where \(\epsilon ^i\) is the ith unit vector in \({\mathbb

We give asymptotically converging semidefinite programming hierarchies of outer bounds on bilinear programs of the form \({\mathrm {Tr}}\big [H(D\otimes E)\big ]\), maximized with respect to semidefinite constraints on D and E. Applied to the problem of approximate error correction in quantum information theory, this gives hierarchies of efficiently computable outer bounds on the success probability

We study the block-coordinate forward–backward algorithm in which the blocks are updated in a random and possibly parallel manner, according to arbitrary probabilities. The algorithm allows different stepsizes along the block-coordinates to fully exploit the smoothness properties of the objective function. In the convex case and in an infinite dimensional setting, we establish almost sure weak convergence

We introduce the integrality number of an integer program (IP). Roughly speaking, the integrality number is the smallest number of integer constraints needed to solve an IP via a mixed integer (MIP) relaxation. One notable property of this number is its invariance under unimodular transformations of the constraint matrix. Considering the largest minor \(\varDelta \) of the constraint matrix, our analysis

We present a connection between two dynamical systems arising in entirely different contexts: the Iteratively Reweighted Least Squares (IRLS) algorithm used in compressed sensing and sparse recovery to find a minimum \(\ell _1\)-norm solution in an affine space, and the dynamics of a slime mold (Physarum polycephalum) that finds the shortest path in a maze. We elucidate this connection by presenting

There has been a recent surge of interest in incorporating fairness aspects into classical clustering problems. Two recently introduced variants of the k-Center problem in this spirit are Colorful k-Center, introduced by Bandyapadhyay, Inamdar, Pai, and Varadarajan, and lottery models, such as the Fair Robust k-Center problem introduced by Harris, Pensyl, Srinivasan, and Trinh. To address fairness

It is well-known that the sum of two quasiconvex functions is not quasiconvex in general, and the same occurs with the minimum. Although apparently these two statements (for the sum or minimum) have nothing in common, they are related, as we show in this paper. To develop our study, the notion of quasiconvex family is introduced, and we establish various characterizations of such a concept: one of

Randomized Kaczmarz, Motzkin Method and Sampling Kaczmarz Motzkin (SKM) algorithms are commonly used iterative techniques for solving a system of linear inequalities (i.e., \(Ax \le b\)). As linear systems of equations represent a modeling paradigm for solving many optimization problems, these randomized and iterative techniques are gaining popularity among researchers in different domains. In this

Given two matroids \(\mathcal {M}_{1} = (E, \mathcal {B}_{1})\) and \(\mathcal {M}_{2} = (E, \mathcal {B}_{2})\) on a common ground set E with base sets \(\mathcal {B}_1\) and \(\mathcal {B}_2\), some integer \(k \in \mathbb {N}\), and two cost functions \(c_{1}, c_{2} :E \rightarrow \mathbb {R}\), we consider the optimization problem to find a basis \(X \in \mathcal {B}_{1}\) and a basis \(Y \in \mathcal