Abstract We address the problem of determining convergent upper bounds in continuous non-convex global minimization of box-constrained problems with equality constraints. These upper bounds are important for the termination of spatial branch-and-bound algorithms. Our method is based on the theorem of Miranda which helps to ensure the existence of feasible points in certain boxes. Then, the computation

Abstract We introduce an inexact variant of stochastic mirror descent (SMD), called inexact stochastic mirror descent (ISMD), to solve nonlinear two-stage stochastic programs where the second stage problem has linear and nonlinear coupling constraints and a nonlinear objective function which depends on both first and second stage decisions. Given a candidate first stage solution and a realization of

Abstract A strategy is proposed for characterizing the worst-case performance of algorithms for solving nonconvex smooth optimization problems. Contemporary analyses characterize worst-case performance by providing, under certain assumptions on an objective function, an upper bound on the number of iterations (or function or derivative evaluations) required until a pth-order stationarity condition

Abstract We consider integer linear programs whose solutions are binary matrices and whose (sub-)symmetry groups are symmetric groups acting on (sub-)columns. Such structured sub-symmetry groups arise in important classes of combinatorial problems, e.g. graph coloring or unit commitment. For a priori known (sub-)symmetries, we propose a framework to build (sub-)symmetry-breaking inequalities for such

Abstract We propose a sequential homotopy method for the solution of mathematical programming problems formulated in abstract Hilbert spaces under the Guignard constraint qualification. The method is equivalent to performing projected backward Euler timestepping on a projected gradient/antigradient flow of the augmented Lagrangian. The projected backward Euler equations can be interpreted as the necessary

Abstract We present a convergence rate analysis for biased stochastic gradient descent (SGD), where individual gradient updates are corrupted by computation errors. We develop stochastic quadratic constraints to formulate a small linear matrix inequality (LMI) whose feasible points lead to convergence bounds of biased SGD. Based on this LMI condition, we develop a sequential minimization approach to

Abstract In this paper, we consider multi-stage stochastic optimization problems with convex objectives and conic constraints at each stage. We present a new stochastic first-order method, namely the dynamic stochastic approximation (DSA) algorithm, for solving these types of stochastic optimization problems. We show that DSA can achieve an optimal \({{\mathcal {O}}}(1/\epsilon ^4)\) rate of convergence

Abstract In this paper, we study the problem of distributed multi-agent optimization over a network, where each agent possesses a local cost function that is smooth and strongly convex. The global objective is to find a common solution that minimizes the average of all cost functions. Assuming agents only have access to unbiased estimates of the gradients of their local cost functions, we consider

Abstract In the multiway cut problem, we are given an undirected graph with non-negative edge weights and a collection of k terminal nodes, and the goal is to partition the node set of the graph into k non-empty parts each containing exactly one terminal, so that the total weight of the edges crossing the partition is minimized. The multiway cut problem for \(k\ge 3\) is APX-hard. For arbitrary k,

Abstract The stochastic Frank–Wolfe method has recently attracted much general interest in the context of optimization for statistical and machine learning due to its ability to work with a more general feasible region. However, there has been a complexity gap in the dependence on the optimality tolerance \(\varepsilon \) in the guaranteed convergence rate for stochastic Frank–Wolfe compared to its

Abstract Sequential optimality conditions have played a major role in unifying and extending global convergence results for several classes of algorithms for general nonlinear optimization. In this paper, we extend theses concepts for nonlinear semidefinite programming. We define two sequential optimality conditions for nonlinear semidefinite programming. The first is a natural extension of the so-called

Abstract A celebrated theorem of Balas gives a linear mixed-integer formulation for the union of two nonempty polytopes whose relaxation gives the convex hull of this union. The number of inequalities in Balas formulation is linear in the number of inequalities that describe the two polytopes and the number of variables is doubled. In this paper we show that this is best possible: in every dimension

Abstract In this paper we study the convergence of an Inertial Forward–Backward algorithm, with a particular choice of an over-relaxation term. In particular we show that for a sequence of over-relaxation parameters, that do not satisfy Nesterov’s rule, one can still expect some relatively fast convergence properties for the objective function. In addition we complement this work by studying the convergence

Abstract The purpose of this manuscript is to derive new convergence results for several subgradient methods applied to minimizing nonsmooth convex functions with Hölderian growth. The growth condition is satisfied in many applications and includes functions with quadratic growth and weakly sharp minima as special cases. To this end there are three main contributions. First, for a constant and sufficiently

Abstract Conditional value at risk (CVaR) has been widely studied as a risk measure. In this paper we add to this work by focusing on the choice of confidence level and its impact on optimization problems with CVaR appearing in the objective and also the constraints. We start by considering a problem in which CVaR is minimized and investigate the way in which it approximates the minimax robust optimization

Abstract In this paper, we present two new methods for solving convex mixed-integer nonlinear programming problems based on the outer approximation method. The first method is inspired by the level method and uses a regularization technique to reduce the step size when choosing new integer combinations. The second method combines ideas from both the level method and the sequential quadratic programming

Abstract Generalizing certain network interdiction games communicated to us by Andrew Liu and his collaborators, this paper studies a bilevel, non-cooperative game wherein the objective function of each player’s optimization problem contains a value function of a second-level linear program parameterized by the first-level variables in a non-convex manner. In the applied network interdiction games

Abstract We consider minimization of a smooth nonconvex objective function using an iterative algorithm based on Newton’s method and the linear conjugate gradient algorithm, with explicit detection and use of negative curvature directions for the Hessian of the objective function. The algorithm tracks Newton-conjugate gradient procedures developed in the 1980s closely, but includes enhancements that

Abstract The paper conducts a second-order variational analysis for an important class of nonpolyhedral conic programs generated by the so-called second-order/Lorentz/ice-cream cone \({\mathcal {Q}}\). From one hand, we prove that the indicator function of \({\mathcal {Q}}\) is always twice epi-differentiable and apply this result to characterizing the uniqueness of Lagrange multipliers together with

Abstract This paper reveals that a common and central role, played in many error bound (EB) conditions and a variety of gradient-type methods, is a residual measure operator. On one hand, by linking this operator with other optimality measures, we define a group of abstract EB conditions, and then analyze the interplay between them; on the other hand, by using this operator as an ascent direction,

Abstract The maximum number of edge-disjoint spanning trees in a network has been used as a measure of the strength of a network. It gives the number of disjoint ways that the network can be fully connected. This suggests a game theoretic analysis that shows the relative importance of the different links to form a strong network. We introduce the Network strength game as a cooperative game defined

Abstract We present a new class of decentralized first-order methods for nonsmooth and stochastic optimization problems defined over multiagent networks. Considering that communication is a major bottleneck in decentralized optimization, our main goal in this paper is to develop algorithmic frameworks which can significantly reduce the number of inter-node communications. Our major contribution is

Abstract We employ chordal decomposition to reformulate a large and sparse semidefinite program (SDP), either in primal or dual standard form, into an equivalent SDP with smaller positive semidefinite (PSD) constraints. In contrast to previous approaches, the decomposed SDP is suitable for the application of first-order operator-splitting methods, enabling the development of efficient and scalable

Abstract We present necessary conditions for monotonicity of fixed point iterations of mappings that may violate the usual nonexpansive property. Notions of linear-type monotonicity of fixed point sequences—weaker than Fejér monotonicity—are shown to imply metric subregularity. This, together with the almost averaging property recently introduced by Luke et al. (Math Oper Res, 2018. https://doi.org/10

Abstract We consider continuous linear programs over a continuous finite time horizon T, with a constant coefficient matrix, linear right hand side functions and linear cost coefficient functions. Specifically, we search for optimal solutions in the space of measures or of functions of bounded variation. These models generalize the separated continuous linear programming models and their various duals

Abstract We provide an efficient algorithm for computing the nucleolus for an instance of a weighted cooperative matching game. This resolves a long-standing open question posed in Faigle (Math Programm, 83: 555–569, 1998).

Abstract We investigate three competing notions that generalize the notion of a facet of finite-dimensional polyhedra to the infinite-dimensional Gomory–Johnson model. These notions were known to coincide for continuous piecewise linear functions with rational breakpoints. We show that two of the notions, extreme functions and facets, coincide for the case of continuous piecewise linear functions,

The theory on the traditional sample average approximation (SAA) scheme for stochastic programming (SP) dictates that the number of samples should be polynomial in the number of problem dimensions in order to ensure proper optimization accuracy. In this paper, we study a modification to the SAA in the scenario where the global minimizer is either sparse or can be approximated by a sparse solution.

The Support Vector Machine (SVM) is one of the most popular classification methods in the machine learning literature. Binary SVM methods have been extensively studied, and have achieved many successes in various disciplines. However, generalization to Multicategory SVM (MSVM) methods can be very challenging. Many existing methods estimate k functions for k classes with an explicit sum-to-zero constraint

In this paper we study optimization problems related to bipartite quantum correlations using techniques from tracial noncommutative polynomial optimization. First we consider the problem of finding the minimal entanglement dimension of such correlations. We construct a hierarchy of semidefinite programming lower bounds and show convergence to a new parameter: the minimal average entanglement dimension

This paper concerns the folded concave penalized sparse linear regression (FCPSLR), a class of popular sparse recovery methods. Although FCPSLR yields desirable recovery performance when solved globally, computing a global solution is NP-complete. Despite some existing statistical performance analyses on local minimizers or on specific FCPSLR-based learning algorithms, it still remains open questions

The little Grothendieck problem consists of maximizing Σ ij Cijxixj for a positive semidef-inite matrix C, over binary variables xi ∈ {±1}. In this paper we focus on a natural generalization of this problem, the little Grothendieck problem over the orthogonal group. Given C ∈ ℝ dn × dn a positive semidefinite matrix, the objective is to maximize [Formula: see text] restricting Oi to take values in

We introduce new optimized first-order methods for smooth unconstrained convex minimization. Drori and Teboulle [5] recently described a numerical method for computing the N-iteration optimal step coefficients in a class of first-order algorithms that includes gradient methods, heavy-ball methods [15], and Nesterov's fast gradient methods [10,12]. However, the numerical method in [5] is computationally

The problem of minimizing a continuously differentiable convex function over an intersection of closed convex sets is ubiquitous in applied mathematics. It is particularly interesting when it is easy to project onto each separate set, but nontrivial to project onto their intersection. Algorithms based on Newton's method such as the interior point method are viable for small to medium-scale problems

This paper derives new algorithms for signomial programming, a generalization of geometric programming. The algorithms are based on a generic principle for optimization called the MM algorithm. In this setting, one can apply the geometric-arithmetic mean inequality and a supporting hyperplane inequality to create a surrogate function with parameters separated. Thus, unconstrained signomial programming

The possibilities of exploiting the special structure of d.c. programs, which consist of optimising the difference of convex functions, are currently more or less limited to variants of the DCA proposed by Pham Dinh Tao and Le Thi Hoai An in 1997. These assume that either the convex or the concave part, or both, are evaluated by one of their subgradients. In this paper we propose an algorithm which

Adjustable robust optimization (ARO) generally produces better worst-case solutions than static robust optimization (RO). However, ARO is computationally more difficult than RO. In this paper, we provide conditions under which the worst-case objective values of ARO and RO problems are equal. We prove that when the uncertainty is constraint-wise, the problem is convex with respect to the adjustable

We study the classical NP -hard problems of finding maximum-size subsets from given sets of k terminal pairs that can be routed via edge-disjoint paths (MaxEDP) or node-disjoint paths (MaxNDP) in a given graph. The approximability of MaxEDP/MaxNDP is currently not well understood; the best known lower bound is 2 Ω ( log n ) , assuming NP ⊈ DTIME ( n O ( log n ) ) . This constitutes a significant gap

We propose a complete-search algorithm for solving a class of non-convex, possibly infinite-dimensional, optimization problems to global optimality. We assume that the optimization variables are in a bounded subset of a Hilbert space, and we determine worst-case run-time bounds for the algorithm under certain regularity conditions of the cost functional and the constraint set. Because these run-time