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  • Convergent upper bounds in global minimization with nonlinear equality constraints
    Math. Program. (IF 3.785) Pub Date : 2020-04-06
    Christian Füllner, Peter Kirst, Oliver Stein

    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

    更新日期:2020-04-06
  • Inexact stochastic mirror descent for two-stage nonlinear stochastic programs
    Math. Program. (IF 3.785) Pub Date : 2020-04-02
    Vincent Guigues

    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

    更新日期:2020-04-03
  • Regional complexity analysis of algorithms for nonconvex smooth optimization
    Math. Program. (IF 3.785) Pub Date : 2020-04-01
    Frank E. Curtis, Daniel P. Robinson

    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

    更新日期:2020-04-01
  • Symmetry-breaking inequalities for ILP with structured sub-symmetry
    Math. Program. (IF 3.785) Pub Date : 2020-03-30
    Pascale Bendotti, Pierre Fouilhoux, Cécile Rottner

    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

    更新日期:2020-03-30
  • A sequential homotopy method for mathematical programming problems
    Math. Program. (IF 3.785) Pub Date : 2020-03-23
    Andreas Potschka, Hans Georg Bock

    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

    更新日期:2020-03-24
  • Analysis of biased stochastic gradient descent using sequential semidefinite programs
    Math. Program. (IF 3.785) Pub Date : 2020-03-23
    Bin Hu, Peter Seiler, Laurent Lessard

    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

    更新日期:2020-03-24
  • Dynamic stochastic approximation for multi-stage stochastic optimization
    Math. Program. (IF 3.785) Pub Date : 2020-03-20
    Guanghui Lan, Zhiqiang Zhou

    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

    更新日期:2020-03-21
  • Distributed stochastic gradient tracking methods
    Math. Program. (IF 3.785) Pub Date : 2020-03-17
    Shi Pu, Angelia Nedić

    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

    更新日期:2020-03-20
  • Improving the integrality gap for multiway cut
    Math. Program. (IF 3.785) Pub Date : 2020-03-06
    Kristóf Bérczi, Karthekeyan Chandrasekaran, Tamás Király, Vivek Madan

    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,

    更新日期:2020-03-20
  • Generalized stochastic Frank–Wolfe algorithm with stochastic “substitute” gradient for structured convex optimization
    Math. Program. (IF 3.785) Pub Date : 2020-03-04
    Haihao Lu, Robert M. Freund

    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

    更新日期:2020-03-20
  • Optimality conditions and global convergence for nonlinear semidefinite programming
    Math. Program. (IF 3.785) Pub Date : 2018-12-07
    Roberto Andreani, Gabriel Haeser, Daiana S. Viana

    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

    更新日期:2020-03-20
  • Balas formulation for the union of polytopes is optimal
    Math. Program. (IF 3.785) Pub Date : 2019-01-19
    Michele Conforti, Marco Di Summa, Yuri Faenza

    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

    更新日期:2020-03-20
  • Convergence rate of inertial Forward–Backward algorithm beyond Nesterov’s rule
    Math. Program. (IF 3.785) Pub Date : 2018-11-12
    Vassilis Apidopoulos, Jean-François Aujol, Charles Dossal

    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

    更新日期:2020-03-20
  • Faster subgradient methods for functions with Hölderian growth
    Math. Program. (IF 3.785) Pub Date : 2019-01-07
    Patrick R. Johnstone, Pierre Moulin

    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

    更新日期:2020-03-20
  • Varying confidence levels for CVaR risk measures and minimax limits
    Math. Program. (IF 3.785) Pub Date : 2019-01-22
    Edward Anderson, Huifu Xu, Dali Zhang

    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

    更新日期:2020-03-20
  • Using regularization and second order information in outer approximation for convex MINLP
    Math. Program. (IF 3.785) Pub Date : 2018-12-18
    Jan Kronqvist, David E. Bernal, Ignacio E. Grossmann

    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

    更新日期:2020-03-20
  • Piecewise affine parameterized value-function based bilevel non-cooperative games
    Math. Program. (IF 3.785) Pub Date : 2018-11-10
    Tianyu Hao, Jong-Shi Pang

    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

    更新日期:2020-03-20
  • A Newton-CG algorithm with complexity guarantees for smooth unconstrained optimization
    Math. Program. (IF 3.785) Pub Date : 2019-01-19
    Clément W. Royer, Michael O’Neill, Stephen J. Wright

    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

    更新日期:2020-03-20
  • Second-order variational analysis in second-order cone programming
    Math. Program. (IF 3.785) Pub Date : 2018-11-03
    Nguyen T. V. Hang, Boris S. Mordukhovich, M. Ebrahim Sarabi

    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

    更新日期:2020-03-20
  • New analysis of linear convergence of gradient-type methods via unifying error bound conditions
    Math. Program. (IF 3.785) Pub Date : 2019-01-04
    Hui Zhang

    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,

    更新日期:2020-03-20
  • Network strength games: the core and the nucleolus
    Math. Program. (IF 3.785) Pub Date : 2018-11-09
    Mourad Baïou, Francisco Barahona

    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

    更新日期:2020-03-20
  • Communication-efficient algorithms for decentralized and stochastic optimization
    Math. Program. (IF 3.785) Pub Date : 2018-12-07
    Guanghui Lan, Soomin Lee, Yi Zhou

    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

    更新日期:2020-03-20
  • Chordal decomposition in operator-splitting methods for sparse semidefinite programs
    Math. Program. (IF 3.785) Pub Date : 2019-02-20
    Yang Zheng, Giovanni Fantuzzi, Antonis Papachristodoulou, Paul Goulart, Andrew Wynn

    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

    更新日期:2020-03-20
  • Necessary conditions for linear convergence of iterated expansive, set-valued mappings
    Math. Program. (IF 3.785) Pub Date : 2018-10-26
    D. Russell Luke, Marc Teboulle, Nguyen H. Thao

    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

    更新日期:2020-03-20
  • A simplex-type algorithm for continuous linear programs with constant coefficients
    Math. Program. (IF 3.785) Pub Date : 2018-12-03
    Evgeny Shindin, Gideon Weiss

    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

    更新日期:2020-03-20
  • Computing the nucleolus of weighted cooperative matching games in polynomial time
    Math. Program. (IF 3.785) Pub Date : 2020-02-28
    Jochen Könemann, Kanstantsin Pashkovich, Justin Toth

    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).

    更新日期:2020-03-20
  • Facets, weak facets, and extreme functions of the Gomory–Johnson infinite group problem
    Math. Program. (IF 3.785) Pub Date : 2020-02-28
    Matthias Köppe, Yuan Zhou

    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,

    更新日期:2020-03-20
  • Sample Average Approximation with Sparsity-Inducing Penalty for High-Dimensional Stochastic Programming.
    Math. Program. (IF 3.785) Pub Date : 2019-11-05
    Hongcheng Liu,Xue Wang,Tao Yao,Runze Li,Yinyu Ye

    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.

    更新日期:2019-11-01
  • Robust Multicategory Support Vector Machines using Difference Convex Algorithm.
    Math. Program. (IF 3.785) Pub Date : 2018-05-08
    Chong Zhang,Minh Pham,Sheng Fu,Yufeng Liu

    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

    更新日期:2019-11-01
  • Bounds on entanglement dimensions and quantum graph parameters via noncommutative polynomial optimization.
    Math. Program. (IF 3.785) Pub Date : 2018-01-01
    Sander Gribling,David de Laat,Monique Laurent

    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

    更新日期:2019-11-01
  • Folded concave penalized sparse linear regression: sparsity, statistical performance, and algorithmic theory for local solutions.
    Math. Program. (IF 3.785) Pub Date : 2017-12-12
    Hongcheng Liu,Tao Yao,Runze Li,Yinyu Ye

    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

    更新日期:2019-11-01
  • Approximating the Little Grothendieck Problem over the Orthogonal and Unitary Groups.
    Math. Program. (IF 3.785) Pub Date : 2016-11-22
    Afonso S Bandeira,Christopher Kennedy,Amit Singer

    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

    更新日期:2019-11-01
  • Optimized first-order methods for smooth convex minimization.
    Math. Program. (IF 3.785) Pub Date : 2016-10-22
    Donghwan Kim,Jeffrey A Fessler

    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

    更新日期:2019-11-01
  • Distance majorization and its applications.
    Math. Program. (IF 3.785) Pub Date : 2014-11-14
    Eric C Chi,Hua Zhou,Kenneth Lange

    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

    更新日期:2019-11-01
  • MM Algorithms for Geometric and Signomial Programming.
    Math. Program. (IF 3.785) Pub Date : 2014-03-19
    Kenneth Lange,Hua Zhou

    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

    更新日期:2019-11-01
  • A general double-proximal gradient algorithm for d.c. programming.
    Math. Program. (IF 3.785) Pub Date : null
    Sebastian Banert,Radu Ioan Boț

    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

    更新日期:2019-11-01
  • When are static and adjustable robust optimization problems with constraint-wise uncertainty equivalent?
    Math. Program. (IF 3.785) Pub Date : null
    Ahmadreza Marandi,Dick den Hertog

    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

    更新日期:2019-11-01
  • New algorithms for maximum disjoint paths based on tree-likeness.
    Math. Program. (IF 3.785) Pub Date : null
    Krzysztof Fleszar,Matthias Mnich,Joachim Spoerhase

    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

    更新日期:2019-11-01
  • Global optimization in Hilbert space.
    Math. Program. (IF 3.785) Pub Date : null
    Boris Houska,Benoît Chachuat

    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

    更新日期:2019-11-01
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