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• Robust inversion, dimensionality reduction, and randomized sampling AI翻译
  作者：Aravkin, Aleksandr; Friedlander, Michael P.; Herrmann, Felix J.; van Leeuwen, Tristan
  作者单位：University of British Columbia; University of British Columbia
  摘要：We consider a class of inverse problems in which the forward model is the solution operator to linear ODEs or PDEs. This class admits several dimensionality-reduction techniques based on data averaging or sampling, which are especially useful for large-scale problems. We survey these approaches and their connection to stochastic optimization. The data-averaging approach is only viable, however, for a least-squares misfit, which is sensitive to outliers in the data and artifacts unexplained by ...
• Computation of sparse low degree interpolating polynomials and their application to derivative-free optimization AI翻译
  作者：Bandeira, A. S.; Scheinberg, K.; Vicente, L. N.
  作者单位：Lehigh University; Princeton University; Universidade de Coimbra
  摘要：Interpolation-based trust-region methods are an important class of algorithms for Derivative-Free Optimization which rely on locally approximating an objective function by quadratic polynomial interpolation models, frequently built from less points than there are basis components. Often, in practical applications, the contribution of the problem variables to the objective function is such that many pairwise correlations between variables are negligible, implying, in the smooth case, a sparse s...
• Optimal control of molecular dynamics using Markov state models AI翻译
  作者：Schuette, Christof; Winkelmann, Stefanie; Hartmann, Carsten
  作者单位：Free University of Berlin
  摘要：A numerical scheme for solving high-dimensional stochastic control problems on an infinite time horizon that appear relevant in the context of molecular dynamics is outlined. The scheme rests on the interpretation of the corresponding Hamilton-Jacobi-Bellman equation as a nonlinear eigenvalue problem that, using a logarithmic transformation, can be recast as a linear eigenvalue problem, for which the principal eigenvalue and its eigenfunction are sought. The latter can be computed efficiently ...
• Global optimization of mixed-integer quadratically-constrained quadratic programs (MIQCQP) through piecewise-linear and edge-concave relaxations AI翻译
  作者：Misener, Ruth; Floudas, Christodoulos A.
  作者单位：Princeton University
  摘要：We propose a deterministic global optimization approach, whose novel contributions are rooted in the edge-concave and piecewise-linear underestimators, to address nonconvex mixed-integer quadratically-constrained quadratic programs (MIQCQP) to epsilon-global optimality. The facets of low-dimensional (n <= 3) edge-concave aggregations dominating the termwise relaxation of MIQCQP are introduced at every node of a branch-and-bound tree. Concave multivariable terms and sparsely distributed bilinea...
• Sample size selection in optimization methods for machine learning AI翻译
  作者：Byrd, Richard H.; Chin, Gillian M.; Nocedal, Jorge; Wu, Yuchen
  作者单位：Northwestern University; University of Colorado System; University of Colorado Boulder; Alphabet Inc.; Google Incorporated
  摘要：This paper presents a methodology for using varying sample sizes in batch-type optimization methods for large-scale machine learning problems. The first part of the paper deals with the delicate issue of dynamic sample selection in the evaluation of the function and gradient. We propose a criterion for increasing the sample size based on variance estimates obtained during the computation of a batch gradient. We establish an complexity bound on the total cost of a gradient method. The second pa...
• SIP: critical value functions have finite modulus of non-convexity AI翻译
  作者：Dorsch, D.; Guerra-Vazquez, F.; Guenzel, H.; Jongen, H. Th.; Rueckmann, J. -J.
  作者单位：University of Birmingham; RWTH Aachen University; Universidad Americas Puebla (UDLAP)
  摘要：We consider semi-infinite programming problems depending on a finite dimensional parameter . Provided that is a strongly stable stationary point of , there exists a locally unique and continuous stationary point mapping . This defines the local critical value function , where denotes the objective function of for a given parameter vector . We show that is the sum of a convex function and a smooth function. In particular, this excludes the appearance of negative kinks in the graph of .
• Semidefinite optimization in discrepancy theory AI翻译
  作者：Bansal, Nikhil
  作者单位：Eindhoven University of Technology
  摘要：Recently, there have been several new developments in discrepancy theory based on connections to semidefinite programming. This connection has been useful in several ways. It gives efficient polynomial time algorithms for several problems for which only non-constructive results were previously known. It also leads to several new structural results in discrepancy itself, such as tightness of the so-called determinant lower bound, improved bounds on the discrepancy of the union of set systems an...
• The adaptive convexification algorithm for semi-infinite programming with arbitrary index sets AI翻译
  作者：Stein, Oliver; Steuermann, Paul
  作者单位：Helmholtz Association; Karlsruhe Institute of Technology
  摘要：A numerical solution method for semi-infinite optimization problems with arbitrary, not necessarily box-shaped, index sets is presented. Following the ideas of Floudas and Stein (SIAM J Optim 18:1187-1208, 2007), convex relaxations of the lower level problem are adaptively constructed and then reformulated as mathematical programs with complementarity constraints and solved. Although the index set is arbitrary, this approximation produces feasible iterates for the original problem. The convex ...
• Smoothing methods for nonsmooth, nonconvex minimization AI翻译
  作者：Chen, Xiaojun
  作者单位：Hong Kong Polytechnic University
  摘要：We consider a class of smoothing methods for minimization problems where the feasible set is convex but the objective function is not convex, not differentiable and perhaps not even locally Lipschitz at the solutions. Such optimization problems arise from wide applications including image restoration, signal reconstruction, variable selection, optimal control, stochastic equilibrium and spherical approximations. In this paper, we focus on smoothing methods for solving such optimization problem...
• A note on the implementation of an interior-point algorithm for nonlinear optimization with inexact step computations AI翻译
  作者：Curtis, Frank E.; Huber, Johannes; Schenk, Olaf; Waechter, Andreas
  作者单位：Northwestern University; Lehigh University; University of Basel; Universita della Svizzera Italiana
  摘要：This paper describes an implementation of an interior-point algorithm for large-scale nonlinear optimization. It is based on the algorithm proposed by Curtis et al. (SIAM J Sci Comput 32:3447-3475, 2010), a method that possesses global convergence guarantees to first-order stationary points with the novel feature that inexact search direction calculations are allowed in order to save computational expense. The implementation follows the proposed algorithm, but includes many practical enhanceme...