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

We study a continuous facility location problem on undirected graphs where all edges have unit length and where the facilities may be positioned on the vertices as well as on interior points of the edges. The goal is to cover the entire graph with a minimum number of facilities with covering range \(\delta >0\). In other words, we want to position as few facilities as possible subject to the condition

Let G be an n-node graph without two disjoint odd cycles. The algorithm of Artmann, Weismantel and Zenklusen (STOC’17) for bimodular integer programs can be used to find a maximum weight stable set in G in strongly polynomial time. Building on structural results characterizing sufficiently connected graphs without two disjoint odd cycles, we construct a size-\(O(n^2)\) extended formulation for the

This paper proposes an accelerated proximal point method for maximally monotone operators. The proof is computer-assisted via the performance estimation problem approach. The proximal point method includes various well-known convex optimization methods, such as the proximal method of multipliers and the alternating direction method of multipliers, and thus the proposed acceleration has wide applications

Choice of a risk measure for quantifying risk of an investment portfolio depends on the decision maker’s risk preference. In this paper, we consider the case when such a preference can be described by a law invariant coherent risk measure but the choice of a specific risk measure is ambiguous. We propose a robust spectral risk approach to address such ambiguity. Differing from Wang and Xu (SIAM J Optim

Minimizing a convex function of a measure with a sparsity-inducing penalty is a typical problem arising, e.g., in sparse spikes deconvolution or two-layer neural networks training. We show that this problem can be solved by discretizing the measure and running non-convex gradient descent on the positions and weights of the particles. For measures on a d-dimensional manifold and under some non-degeneracy

In this paper, a novel stochastic extra-step quasi-Newton method is developed to solve a class of nonsmooth nonconvex composite optimization problems. We assume that the gradient of the smooth part of the objective function can only be approximated by stochastic oracles. The proposed method combines general stochastic higher order steps derived from an underlying proximal type fixed-point equation

We study a 1-dimensional discrete signal denoising problem that consists of minimizing a sum of separable convex fidelity terms and convex regularization terms, the latter penalize the differences of adjacent signal values. This problem generalizes the total variation regularization problem. We provide here a unified approach to solve the problem for general convex fidelity and regularization functions

A new primal-dual interior-point algorithm applicable to nonsymmetric conic optimization is proposed. It is a generalization of the famous algorithm suggested by Nesterov and Todd for the symmetric conic case, and uses primal-dual scalings for nonsymmetric cones proposed by Tunçel. We specialize Tunçel’s primal-dual scalings for the important case of 3 dimensional exponential-cones, resulting in a

In the Euclidean setting the proximal gradient method and its accelerated variants are a class of efficient algorithms for optimization problems with decomposable objective. In this paper, we develop a Riemannian proximal gradient method (RPG) and its accelerated variant (ARPG) for similar problems but constrained on a manifold. The global convergence of RPG is established under mild assumptions, and

In a first contribution, we revisit two certificates of positivity on (possibly non-compact) basic semialgebraic sets due to Putinar and Vasilescu (C R Acad Sci Ser I Math 328(6):495–499, 1999). We use Jacobi’s technique from (Math Z 237(2):259–273, 2001) to provide an alternative proof with an effective degree bound on the sums of squares weights in such certificates. As a consequence, it allows one

We present a general numerical solution method for control problems with state variables defined by a linear PDE over a finite set of binary or continuous control variables. We show empirically that a naive approach that applies a numerical discretization scheme to the PDEs to derive constraints for a mixed-integer linear program (MILP) leads to systems that are too large to be solved with state-of-the-art

We present an algorithm for the minimization of a nonconvex quadratic function subject to linear inequality constraints and a two-sided bound on the 2-norm of its solution. The algorithm minimizes the objective using an active-set method by solving a series of trust-region subproblems (TRS). Underpinning the efficiency of this approach is that the global solution of the TRS has been widely studied

We propose finitely convergent methods for solving convex feasibility problems defined over a possibly infinite pool of constraints. Following other works in this area, we assume that the interior of the solution set is nonempty and that certain overrelaxation parameters form a divergent series. We combine our methods with a very general class of deterministic control sequences where, roughly speaking

We present a method to solve two-stage stochastic linear programming problems with fixed recourse when the uncertainty space can have either discrete or continuous distributions. Given a partition of the uncertainty space, the method is addressed to solve a discrete problem with one scenario for each element of the partition (subregions of the uncertainty space). Fixing first-stage variables, we formulate

In a bottleneck combinatorial problem, the objective is to minimize the highest cost of elements of a subset selected from the combinatorial solution space. This paper studies data-driven distributionally robust bottleneck combinatorial problems (DRBCP) with stochastic costs, where the probability distribution of the cost vector is contained in a ball of distributions centered at the empirical distribution

For two matroids \(M_1\) and \(M_2\) with the same ground set V and two cost functions \(w_1\) and \(w_2\) on \(2^V\), we consider the problem of finding bases \(X_1\) of \(M_1\) and \(X_2\) of \(M_2\) minimizing \(w_1(X_1)+w_2(X_2)\) subject to a certain cardinality constraint on their intersection \(X_1 \cap X_2\). For this problem, Lendl et al. (Matroid bases with cardinality constraints on the

For a set X of integer points in a polyhedron, the smallest number of facets of any polyhedron whose set of integer points coincides with X is called the relaxation complexity \({{\,\mathrm{rc}\,}}(X)\). This parameter, introduced by Kaibel & Weltge (2015), captures the complexity of linear descriptions of X without using auxiliary variables. Using tools from combinatorics, geometry of numbers, and

We study the equilibrium computation problem for two classical resource allocation games: atomic splittable congestion games and multimarket Cournot oligopolies. For atomic splittable congestion games with singleton strategies and player-specific affine cost functions, we devise the first polynomial time algorithm computing a pure Nash equilibrium. Our algorithm is combinatorial and computes the exact

We propose a dynamic version of the classical node packing problem, also called the stable set or independent set problem. The problem is defined by a node set, a node weight vector, and an edge probability vector. For every pair of nodes, an edge is present or not according to an independent Bernoulli random variable defined by the corresponding entry in the probability vector. At each step, the decision

The network reconfiguration problem seeks to find a rooted tree T such that the energy of the (unique) feasible electrical flow over T is minimized. The tree requirement on the support of the flow is motivated by operational constraints in electricity distribution networks. The bulk of existing results on convex optimization over vertices of polytopes and on the structure of electrical flows do not

We study the local convergence of classical quasi-Newton methods for nonlinear optimization. Although it was well established a long time ago that asymptotically these methods converge superlinearly, the corresponding rates of convergence still remain unknown. In this paper, we address this problem. We obtain first explicit non-asymptotic rates of superlinear convergence for the standard quasi-Newton

Bilevel optimization problems have received a lot of attention in the last years and decades. Besides numerous theoretical developments there also evolved novel solution algorithms for mixed-integer linear bilevel problems and the most recent algorithms use branch-and-cut techniques from mixed-integer programming that are especially tailored for the bilevel context. In this paper, we consider MIQP-QP

In this paper, we develop an algorithm to efficiently solve risk-averse optimization problems posed in reflexive Banach space. Such problems often arise in many practical applications as, e.g., optimization problems constrained by partial differential equations with uncertain inputs. Unfortunately, for many popular risk models including the coherent risk measures, the resulting risk-averse objective

We consider exact deterministic mixed-integer programming (MIP) reformulations of distributionally robust chance-constrained programs (DR-CCP) with random right-hand sides over Wasserstein ambiguity sets. The existing MIP formulations are known to have weak continuous relaxation bounds, and, consequently, for hard instances with small radius, or with large problem sizes, the branch-and-bound based

We consider a problem of maximizing a monotone DR-submodular function under multiple order-consistent knapsack constraints on a distributive lattice. Because a distributive lattice is used to represent a dependency constraint, the problem can represent a dependency constrained version of a submodular maximization problem on a set. We propose a (\(1 - 1/e\))-approximation algorithm for this problem

A displacement aggregation strategy is proposed for the curvature pairs stored in a limited-memory BFGS (a.k.a. L-BFGS) method such that the resulting (inverse) Hessian approximations are equal to those that would be derived from a full-memory BFGS method. This means that, if a sufficiently large number of pairs are stored, then an optimization algorithm employing the limited-memory method can achieve

This article focuses on optimization of polynomials in noncommuting variables, while taking into account sparsity in the input data. A converging hierarchy of semidefinite relaxations for eigenvalue and trace optimization is provided. This hierarchy is a noncommutative analogue of results due to Lasserre (SIAM J Optim 17(3):822–843, 2006) and Waki et al. (SIAM J Optim 17(1):218–242, 2006). The Gelfand–Naimark–Segal

Using polarity, we give an outer polyhedral approximation for the epigraph of set functions. For a submodular function, we prove that the corresponding polar relaxation is exact; hence, it is equivalent to the Lovász extension. The polar approach provides an alternative proof for the convex hull description of the epigraph of a submodular function. Computational experiments show that the inequalities

We study a robust monopoly pricing problem with a minimax regret objective, where a seller endeavors to sell multiple goods to a single buyer, only knowing that the buyer’s values for the goods range over a rectangular uncertainty set. We interpret this pricing problem as a zero-sum game between the seller, who chooses a selling mechanism, and a fictitious adversary or ‘nature’, who chooses the buyer’s

We study the convex relaxation of a polynomial optimization problem, maximizing a product of linear forms over the complex sphere. We show that this convex program is also a relaxation of the permanent of Hermitian positive semidefinite (HPSD) matrices. By analyzing a constructive randomized rounding algorithm, we obtain an improved multiplicative approximation factor to the permanent of HPSD matrices

The problem of minimizing a (nonconvex) quadratic form over the unit simplex, referred to as a standard quadratic program, admits an exact convex conic formulation over the computationally intractable cone of completely positive matrices. Replacing the intractable cone in this formulation by the larger but tractable cone of doubly nonnegative matrices, i.e., the cone of positive semidefinite and componentwise

We consider nonlinear multistage stochastic optimization problems in the spaces of integrable functions. We allow for nonlinear dynamics and general objective functionals, including dynamic risk measures. We study causal operators describing the dynamics of the system and derive the Clarke subdifferential for a penalty function involving such operators. Then we introduce the concept of subregular recourse

This paper analyzes block-coordinate proximal gradient methods for minimizing the sum of a separable smooth function and a (nonseparable) nonsmooth function, both of which are allowed to be nonconvex. The main tool in our analysis is the forward-backward envelope, which serves as a particularly suitable continuous and real-valued Lyapunov function. Global and linear convergence results are established

In a Hilbert space \( {\mathcal H}\), we introduce a new class of first-order algorithms which naturally occur as discrete temporal versions of an inertial differential inclusion jointly involving viscous friction and dry friction. The function \(f:{\mathcal H}\rightarrow {\mathbb {R}}\) to be minimized is supposed to be differentiable (not necessarily convex), and enters the algorithm via its gradient

The Nemhauser–Trotter theorem states that the standard linear programming (LP) formulation for the stable set problem has a remarkable property, also known as (weak) persistency: for every optimal LP solution that assigns integer values to some variables, there exists an optimal integer solution in which these variables retain the same values. While the standard LP is defined by only non-negativity

In this paper, we consider the subcontracting and single-item lot-sizing problem with constant capacities, which is uncapacitated in subcontracting but capacitated in production. For a holistic understanding of the problem, an infinite-period model is proposed. Such a model provides a unified view of a capacitated lot-sizing problem. The usefulness of the infinite-period model is shown by the principle

Quadratically constrained quadratic programs (QCQPs) are a fundamental class of optimization problems well-known to be NP-hard in general. In this paper we study conditions under which the standard semidefinite program (SDP) relaxation of a QCQP is tight. We begin by outlining a general framework for proving such sufficient conditions. Then using this framework, we show that the SDP relaxation is tight

In this paper, we study local convergence of high-order Tensor Methods for solving convex optimization problems with composite objective. We justify local superlinear convergence under the assumption of uniform convexity of the smooth component, having Lipschitz-continuous high-order derivative. The convergence both in function value and in the norm of minimal subgradient is established. Global complexity

We introduce a new approach to develop stochastic optimization algorithms for a class of stochastic composite and possibly nonconvex optimization problems. The main idea is to combine a variance-reduced estimator and an unbiased stochastic one to create a new hybrid estimator which trades-off the variance and bias, and possesses useful properties for developing new algorithms. We first introduce our

The maximum-entropy sampling problem is a fundamental and challenging combinatorial-optimization problem, with application in spatial statistics. It asks to find a maximum-determinant order-s principal submatrix of an order-n covariance matrix. Exact solution methods for this NP-hard problem are based on a branch-and-bound framework. Many of the known upper bounds for the optimal value are based on

In this paper we establish general formulas for the subdifferential of the pointwise supremum of convex functions, which cover and unify both the compact continuous and the non-compact non-continuous settings. From the non-continuous to the continuous setting, we proceed by a compactification-based approach which leads us to problems having compact index sets and upper semi-continuously indexed mappings

The combinatorial integral approximation decomposition splits the optimization of a discrete-valued control into two steps: solving a continuous relaxation of the discrete control problem, and computing a discrete-valued approximation of the relaxed control. Different algorithms exist for the second step to construct piecewise constant discrete-valued approximants that are defined on given decompositions

We propose a doubly nonnegative (DNN) relaxation for polynomial optimization problems (POPs) with binary and box constraints. This work is an extension of the work by Kim, Kojima and Toh in 2016 from quadratic optimization problems to POPs. The dense and sparse DNN relaxations are reduced to a simple conic optimization problem (COP) to which an accelerated bisection and projection (BP) algorithm is

In this work, we study the optimization landscape of the non-convex matrix sensing problem that is known to have many local minima in the worst case. Since the existing results are related to the notion of restricted isometry property (RIP) that cannot directly capture the underlying structure of a given problem, they can hardly be applied to real-world problems where the amount of data is not exorbitantly

The paper provides a connection between Commutative Algebra and Integer Programming and contains two parts. The first one is devoted to the asymptotic behavior of integer programs with a fixed cost linear functional and the constraint sets consisting of a finite system of linear equations or inequalities with integer coefficients depending linearly on n. An integer \(N_*\) is determined such that the

In this paper, we introduce some new notions of quasi efficiency and quasi proper efficiency for multiobjective optimization problems that reduce to the most important concepts of approximate and quasi efficient solutions given up to now. We establish main properties and provide characterizations for these solutions by linear and nonlinear scalarizations. With the help of quasi efficient solutions

Quantum states are represented by positive semidefinite Hermitian operators with unit trace, known as density matrices. An important subset of quantum states is that of separable states, the complement of which is the subset of entangled states. We show that the problem of deciding whether a quantum state is entangled can be seen as a moment problem in real analysis. Only a small number of such moments

In a Hilbert space setting, for convex optimization, we analyze the convergence rate of a class of first-order algorithms involving inertial features. They can be interpreted as discrete time versions of inertial dynamics involving both viscous and Hessian-driven dampings. The geometrical damping driven by the Hessian intervenes in the dynamics in the form \(\nabla ^2 f (x(t)) \dot{x} (t)\). By treating

The paper investigates analytical properties of dynamic probabilistic constraints (chance constraints). The underlying random distribution is supposed to be continuous. In the first part, a general multistage model with decision rules depending on past observations of the random process is analyzed. Basic properties like (weak sequential) (semi-) continuity of the probability function or existence

A clutter is k-wise intersecting if every k members have a common element, yet no element belongs to all members. We conjecture that, for some integer \(k\ge 4\), every k-wise intersecting clutter is non-ideal. As evidence for our conjecture, we prove it for \(k=4\) for the class of binary clutters. Two key ingredients for our proof are Jaeger’s 8-flow theorem for graphs, and Seymour’s characterization

We show that, if \({\varvec{W}}\) is an \(N \times N\) matrix drawn from the gaussian orthogonal ensemble, then with high probability the degree 4 sum-of-squares relaxation cannot certify an upper bound on the objective \(N^{-1} \cdot \varvec{x}^\top \varvec{W} \varvec{x}\) under the constraints \(x_i^2 - 1 = 0\) (i.e. \(\varvec{x}\in \{\pm 1 \}^N\)) that is asymptotically smaller than \(\lambda _{\max

We show that the Euclidean ball has the smallest volume among sublevel sets of nonnegative forms of bounded Bombieri norm as well as among sublevel sets of sum of squares forms whose Gram matrix has bounded Frobenius or nuclear (or, more generally, p-Schatten) norm. These volume-minimizing properties of the Euclidean ball with respect to its representation (as a sublevel set of a form of fixed even

Smoothed analysis is a powerful paradigm in overcoming worst-case intractability in high-dimensional data analysis and unsupervised learning. While polynomial time smoothed analysis guarantees have been obtained for worst-case intractable problems like tensor decomposition and learning mixtures of Gaussians, such guarantees have been hard to obtain for several other important problems in data analysis

We consider a hierarchy of upper approximations for the minimization of a polynomial f over a compact set \(K \subseteq \mathbb {R}^n\) proposed recently by Lasserre (arXiv:1907.097784, 2019). This hierarchy relies on using the push-forward measure of the Lebesgue measure on K by the polynomial f and involves univariate sums of squares of polynomials with growing degrees 2r. Hence it is weaker, but