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作者：Patel, Vivak
作者单位：University of Wisconsin System; University of Wisconsin Madison
摘要：Stopping criteria for Stochastic Gradient Descent (SGD) methods play important roles from enabling adaptive step size schemes to providing rigor for downstream analyses such as asymptotic inference. Unfortunately, current stopping criteria for SGD methods are often heuristics that rely on asymptotic normality results or convergence to stationary distributions, which may fail to exist for nonconvex functions and, thereby, limit the applicability of such stopping criteria. To address this issue,...
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作者：Aliev, Iskander; Averkov, Gennadiy; De Loera, Jesus A.; Oertel, Timm
作者单位：Cardiff University; Brandenburg University of Technology Cottbus; University of California System; University of California Davis
摘要：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 l(0)-norm of the vector. Our main results are new improved bounds on the minimal l(0)-norm of solutions to systems Ax = b, where A is an element of Z(mxn), b is an element of Z(m) and x is either a general integer vector (lattice case) or a non-negative integer vector (s...
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作者：Averkov, Gennadiy; Schymura, Matthias
作者单位：Brandenburg University of Technology Cottbus
摘要：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 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 quantifier elimination, we make progress on several open questions regarding rc(X) and its variant rcQ(X), restrictin...
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作者：Bienstock, Daniel; Zambelli, Giacomo
作者单位：Columbia University; University of London; London School Economics & Political Science
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作者：Chen, Rui; Luedtke, James
作者单位：University of Wisconsin System; University of Wisconsin Madison
摘要：We investigate sample average approximation (SAA) for two-stage stochastic programs without relatively complete recourse, i.e., for problems in which there are first-stage feasible solutions that are not guaranteed to have a feasible recourse action. As a feasibility measure of the SAA solution, we consider the recourse likelihood, which is the probability that the solution has a feasible recourse action. For epsilon is an element of (0, 1), we demonstrate that the probability that a SAA solut...
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作者：Bauschke, Heinz H.; Ouyang, Hui; Wang, Xianfu
作者单位：University of British Columbia
摘要：The notion of best approximation mapping (BAM) with respect to a closed affine subspace in finite-dimensional spaces was introduced by Behling, Bello Cruz and Santos to show the linear convergence of the block-wise circumcentered-reflection method. The best approximation mapping possesses two critical properties of the circumcenter mapping for linear convergence. Because the iteration sequence of a BAM linearly converges, the BAM is interesting in its own right. In this paper, we naturally ext...
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作者：Gannot, Oran
摘要：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 the Triple Momentum Method. To further emphasize the usefulness of frequency-domain methods, we d...
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作者：Guo, Shaoyan; Xu, Huifu
作者单位：Dalian University of Technology; Chinese University of Hong Kong
摘要：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 30(4):3198-3229, 2020), the new robust model allows one to elicit the decision maker's risk ...
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作者：Ramirez-Pico, Cristian; Moreno, Eduardo
作者单位：Universidad Adolfo Ibanez
摘要：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 a second-stage subproblem for each element, and exploiting information from the dual of t...
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作者：Gonzalez Grandon, T.; Henrion, R.; Perez-Aros, P.
作者单位：Humboldt University of Berlin; Leibniz Association; Weierstrass Institute for Applied Analysis & Stochastics; Universidad de O'Higgins
摘要：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 of solutions are studied. It turns out that the results differ significantly according to whether...
