Pub Date : 2023-10-27DOI: 10.1007/s10589-023-00539-3
Patrick R. Johnstone, Jonathan Eckstein, Thomas Flynn, Shinjae Yoo
{"title":"Correction to: Stochastic projective splitting","authors":"Patrick R. Johnstone, Jonathan Eckstein, Thomas Flynn, Shinjae Yoo","doi":"10.1007/s10589-023-00539-3","DOIUrl":"https://doi.org/10.1007/s10589-023-00539-3","url":null,"abstract":"","PeriodicalId":55227,"journal":{"name":"Computational Optimization and Applications","volume":"40 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136262592","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-25DOI: 10.1007/s10589-023-00536-6
Mikhail A. Karapetyants
Abstract In this paper we would like to address the classical optimization problem of minimizing a proper, convex and lower semicontinuous function via the second order in time dynamics, combining viscous and Hessian-driven damping with a Tikhonov regularization term. In our analysis we heavily exploit the Moreau envelope of the objective function and its properties as well as Tikhonov regularization properties, which we extend to a nonsmooth case. We introduce the setting, which at the same time guarantees the fast convergence of the function (and Moreau envelope) values and strong convergence of the trajectories of the system to a minimal norm solution—the element of the minimal norm of all the minimizers of the objective. Moreover, we deduce the precise rates of convergence of the values for the particular choice of parameters. Various numerical examples are also included as an illustration of the theoretical results.
{"title":"A fast continuous time approach for non-smooth convex optimization using Tikhonov regularization technique","authors":"Mikhail A. Karapetyants","doi":"10.1007/s10589-023-00536-6","DOIUrl":"https://doi.org/10.1007/s10589-023-00536-6","url":null,"abstract":"Abstract In this paper we would like to address the classical optimization problem of minimizing a proper, convex and lower semicontinuous function via the second order in time dynamics, combining viscous and Hessian-driven damping with a Tikhonov regularization term. In our analysis we heavily exploit the Moreau envelope of the objective function and its properties as well as Tikhonov regularization properties, which we extend to a nonsmooth case. We introduce the setting, which at the same time guarantees the fast convergence of the function (and Moreau envelope) values and strong convergence of the trajectories of the system to a minimal norm solution—the element of the minimal norm of all the minimizers of the objective. Moreover, we deduce the precise rates of convergence of the values for the particular choice of parameters. Various numerical examples are also included as an illustration of the theoretical results.","PeriodicalId":55227,"journal":{"name":"Computational Optimization and Applications","volume":"2 5","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135218818","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-20DOI: 10.1007/s10589-023-00535-7
C. Yalçın Kaya, Helmut Maurer
Abstract Simultaneous optimization of multiple objective functions results in a set of trade-off, or Pareto, solutions. Choosing a, in some sense, best solution in this set is in general a challenging task: In the case of three or more objectives the Pareto front is usually difficult to view, if not impossible, and even in the case of just two objectives constructing the whole Pareto front so as to visually inspect it might be very costly. Therefore, optimization over the Pareto (or efficient) set has been an active area of research. Although there is a wealth of literature involving finite dimensional optimization problems in this area, there is a lack of problem formulation and numerical methods for optimal control problems, except for the convex case. In this paper, we formulate the problem of optimizing over the Pareto front of nonconvex constrained and time-delayed optimal control problems as a bi-level optimization problem. Motivated by existing solution differentiability results, we propose an algorithm incorporating (i) the Chebyshev scalarization, (ii) a concept of the essential interval of weights, and (iii) the simple but effective bisection method, for optimal control problems with two objectives. We illustrate the working of the algorithm on two example problems involving an electric circuit and treatment of tuberculosis and discuss future lines of research for new computational methods.
{"title":"Optimization over the Pareto front of nonconvex multi-objective optimal control problems","authors":"C. Yalçın Kaya, Helmut Maurer","doi":"10.1007/s10589-023-00535-7","DOIUrl":"https://doi.org/10.1007/s10589-023-00535-7","url":null,"abstract":"Abstract Simultaneous optimization of multiple objective functions results in a set of trade-off, or Pareto, solutions. Choosing a, in some sense, best solution in this set is in general a challenging task: In the case of three or more objectives the Pareto front is usually difficult to view, if not impossible, and even in the case of just two objectives constructing the whole Pareto front so as to visually inspect it might be very costly. Therefore, optimization over the Pareto (or efficient) set has been an active area of research. Although there is a wealth of literature involving finite dimensional optimization problems in this area, there is a lack of problem formulation and numerical methods for optimal control problems, except for the convex case. In this paper, we formulate the problem of optimizing over the Pareto front of nonconvex constrained and time-delayed optimal control problems as a bi-level optimization problem. Motivated by existing solution differentiability results, we propose an algorithm incorporating (i) the Chebyshev scalarization, (ii) a concept of the essential interval of weights, and (iii) the simple but effective bisection method, for optimal control problems with two objectives. We illustrate the working of the algorithm on two example problems involving an electric circuit and treatment of tuberculosis and discuss future lines of research for new computational methods.","PeriodicalId":55227,"journal":{"name":"Computational Optimization and Applications","volume":"27 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135513761","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-16DOI: 10.1007/s10589-023-00534-8
M. Locatelli
Abstract The convex envelope value for a given function f over a region X at some point $$textbf{x}in X$$ x∈X can be derived by searching for the largest value at that point among affine underestimators of f over X . This can be computed by solving a maximin problem, whose exact computation, however, may be a hard task. In this paper we show that by relaxation of the inner minimization problem, duality, and, in particular, by an enlargement of the class of underestimators (thus, not only affine ones) an easier derivation of good convex understimating functions, which can also be proved to be convex envelopes in some cases, is possible. The proposed approach is mainly applied to the derivation of convex underestimators (in fact, in some cases, convex envelopes) in the quadratic case. However, some results are also presented for polynomial, ratio of polynomials, and some other separable functions over regions defined by similarly defined separable functions.
{"title":"A new technique to derive tight convex underestimators (sometimes envelopes)","authors":"M. Locatelli","doi":"10.1007/s10589-023-00534-8","DOIUrl":"https://doi.org/10.1007/s10589-023-00534-8","url":null,"abstract":"Abstract The convex envelope value for a given function f over a region X at some point $$textbf{x}in X$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>X</mml:mi> </mml:mrow> </mml:math> can be derived by searching for the largest value at that point among affine underestimators of f over X . This can be computed by solving a maximin problem, whose exact computation, however, may be a hard task. In this paper we show that by relaxation of the inner minimization problem, duality, and, in particular, by an enlargement of the class of underestimators (thus, not only affine ones) an easier derivation of good convex understimating functions, which can also be proved to be convex envelopes in some cases, is possible. The proposed approach is mainly applied to the derivation of convex underestimators (in fact, in some cases, convex envelopes) in the quadratic case. However, some results are also presented for polynomial, ratio of polynomials, and some other separable functions over regions defined by similarly defined separable functions.","PeriodicalId":55227,"journal":{"name":"Computational Optimization and Applications","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136079577","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-11DOI: 10.1007/s10589-023-00532-w
Jeffrey Christiansen, Brian Dandurand, Andrew Eberhard, Fabricio Oliveira
Abstract Motivated by recent literature demonstrating the surprising effectiveness of the heuristic application of progressive hedging (PH) to stochastic mixed-integer programming (SMIP) problems, we provide theoretical support for the inclusion of integer variables, bridging the gap between theory and practice. We provide greater insight into the following observed phenomena of PH as applied to SMIP where optimal or at least feasible convergence is observed. We provide an analysis of a modified PH algorithm from a different viewpoint, drawing on the interleaving of (split) proximal-point methods (including PH), Gauss–Seidel methods, and the utilisation of variational analysis tools. Through this analysis, we show that under mild conditions, convergence to a feasible solution should be expected. In terms of convergence analysis, we provide two main contributions. First, we contribute insight into the convergence of proximal-point-like methods in the presence of integer variables via the introduction of the notion of persistent local minima. Secondly, we contribute an enhanced Gauss–Seidel convergence analysis that accommodates the variation of the objective function under mild assumptions. We provide a practical implementation of a modified PH and demonstrate its convergent behaviour with computational experiments in line with the provided analysis.
{"title":"A study of progressive hedging for stochastic integer programming","authors":"Jeffrey Christiansen, Brian Dandurand, Andrew Eberhard, Fabricio Oliveira","doi":"10.1007/s10589-023-00532-w","DOIUrl":"https://doi.org/10.1007/s10589-023-00532-w","url":null,"abstract":"Abstract Motivated by recent literature demonstrating the surprising effectiveness of the heuristic application of progressive hedging (PH) to stochastic mixed-integer programming (SMIP) problems, we provide theoretical support for the inclusion of integer variables, bridging the gap between theory and practice. We provide greater insight into the following observed phenomena of PH as applied to SMIP where optimal or at least feasible convergence is observed. We provide an analysis of a modified PH algorithm from a different viewpoint, drawing on the interleaving of (split) proximal-point methods (including PH), Gauss–Seidel methods, and the utilisation of variational analysis tools. Through this analysis, we show that under mild conditions, convergence to a feasible solution should be expected. In terms of convergence analysis, we provide two main contributions. First, we contribute insight into the convergence of proximal-point-like methods in the presence of integer variables via the introduction of the notion of persistent local minima. Secondly, we contribute an enhanced Gauss–Seidel convergence analysis that accommodates the variation of the objective function under mild assumptions. We provide a practical implementation of a modified PH and demonstrate its convergent behaviour with computational experiments in line with the provided analysis.","PeriodicalId":55227,"journal":{"name":"Computational Optimization and Applications","volume":"30 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136057615","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-10DOI: 10.1007/s10589-023-00533-9
Yassine Nabou, Ion Necoara
{"title":"Efficiency of higher-order algorithms for minimizing composite functions","authors":"Yassine Nabou, Ion Necoara","doi":"10.1007/s10589-023-00533-9","DOIUrl":"https://doi.org/10.1007/s10589-023-00533-9","url":null,"abstract":"","PeriodicalId":55227,"journal":{"name":"Computational Optimization and Applications","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136295687","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-04DOI: 10.1007/s10589-023-00531-x
Constantine A. Vitt, Darinka Dentcheva, Andrzej Ruszczyński, Nolan Sandberg
{"title":"The deepest event cuts in risk-averse optimization with application to radiation therapy design","authors":"Constantine A. Vitt, Darinka Dentcheva, Andrzej Ruszczyński, Nolan Sandberg","doi":"10.1007/s10589-023-00531-x","DOIUrl":"https://doi.org/10.1007/s10589-023-00531-x","url":null,"abstract":"","PeriodicalId":55227,"journal":{"name":"Computational Optimization and Applications","volume":"78 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135548090","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-30DOI: 10.1007/s10589-023-00530-y
William W. Hager
{"title":"Extension of switch point algorithm to boundary-value problems","authors":"William W. Hager","doi":"10.1007/s10589-023-00530-y","DOIUrl":"https://doi.org/10.1007/s10589-023-00530-y","url":null,"abstract":"","PeriodicalId":55227,"journal":{"name":"Computational Optimization and Applications","volume":"30 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136248801","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-28DOI: 10.1007/s10589-023-00527-7
Ensio Suonperä, Tuomo Valkonen
Abstract We propose a new approach to solving bilevel optimization problems, intermediate between solving full-system optimality conditions with a Newton-type approach, and treating the inner problem as an implicit function. The overall idea is to solve the full-system optimality conditions, but to precondition them to alternate between taking steps of simple conventional methods for the inner problem, the adjoint equation, and the outer problem. While the inner objective has to be smooth, the outer objective may be nonsmooth subject to a prox-contractivity condition. We prove linear convergence of the approach for combinations of gradient descent and forward-backward splitting with exact and inexact solution of the adjoint equation. We demonstrate good performance on learning the regularization parameter for anisotropic total variation image denoising, and the convolution kernel for image deconvolution.
{"title":"Linearly convergent bilevel optimization with single-step inner methods","authors":"Ensio Suonperä, Tuomo Valkonen","doi":"10.1007/s10589-023-00527-7","DOIUrl":"https://doi.org/10.1007/s10589-023-00527-7","url":null,"abstract":"Abstract We propose a new approach to solving bilevel optimization problems, intermediate between solving full-system optimality conditions with a Newton-type approach, and treating the inner problem as an implicit function. The overall idea is to solve the full-system optimality conditions, but to precondition them to alternate between taking steps of simple conventional methods for the inner problem, the adjoint equation, and the outer problem. While the inner objective has to be smooth, the outer objective may be nonsmooth subject to a prox-contractivity condition. We prove linear convergence of the approach for combinations of gradient descent and forward-backward splitting with exact and inexact solution of the adjoint equation. We demonstrate good performance on learning the regularization parameter for anisotropic total variation image denoising, and the convolution kernel for image deconvolution.","PeriodicalId":55227,"journal":{"name":"Computational Optimization and Applications","volume":"60 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135386223","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-25DOI: 10.1007/s10589-023-00524-w
Stuart Harwood, Francisco Trespalacios, Dimitri Papageorgiou, Kevin Furman
{"title":"Equilibrium modeling and solution approaches inspired by nonconvex bilevel programming","authors":"Stuart Harwood, Francisco Trespalacios, Dimitri Papageorgiou, Kevin Furman","doi":"10.1007/s10589-023-00524-w","DOIUrl":"https://doi.org/10.1007/s10589-023-00524-w","url":null,"abstract":"","PeriodicalId":55227,"journal":{"name":"Computational Optimization and Applications","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135768746","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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