Nam Ho-Nguyen, F. Kılınç-Karzan, Simge Küçükyavuz, Dabeen Lee
Distributionally robust chance-constrained programs (DR-CCPs) over Wasserstein ambiguity sets exhibit attractive out-of-sample performance and admit big-M–based mixed-integer programming reformulations with conic constraints. However, the resulting formulations often suffer from scalability issues as problem size increases. To address this shortcoming, we derive stronger formulations that scale well with respect to the problem size. Our focus is on ambiguity sets under the so-called left-hand side uncertainty, where the uncertain parameters affect the coefficients of the decision variables in the linear inequalities defining the safety sets. The interaction between the uncertain parameters and the variable coefficients in the safety set definition causes challenges in strengthening the original big-M formulations. By exploiting the connection between nominal chance-constrained programs and DR-CCP, we obtain strong formulations with significant enhancements. In particular, through this connection, we derive a linear number of valid inequalities, which can be immediately added to the formulations to obtain improved formulations in the original space of variables. In addition, we suggest a quantile-based strengthening procedure that allows us to reduce the big-M coefficients drastically. Furthermore, based on this procedure, we propose an exponential class of inequalities that can be separated efficiently within a branch-and-cut framework. The quantile-based strengthening procedure can be expensive. Therefore, for the special case of covering and packing type problems, we identify an efficient scheme to carry out this procedure. We demonstrate the computational efficacy of our proposed formulations on two classes of problems, namely stochastic portfolio optimization and resource planning.
{"title":"Strong Formulations for Distributionally Robust Chance-Constrained Programs with Left-Hand Side Uncertainty Under Wasserstein Ambiguity","authors":"Nam Ho-Nguyen, F. Kılınç-Karzan, Simge Küçükyavuz, Dabeen Lee","doi":"10.1287/ijoo.2022.0083","DOIUrl":"https://doi.org/10.1287/ijoo.2022.0083","url":null,"abstract":"Distributionally robust chance-constrained programs (DR-CCPs) over Wasserstein ambiguity sets exhibit attractive out-of-sample performance and admit big-M–based mixed-integer programming reformulations with conic constraints. However, the resulting formulations often suffer from scalability issues as problem size increases. To address this shortcoming, we derive stronger formulations that scale well with respect to the problem size. Our focus is on ambiguity sets under the so-called left-hand side uncertainty, where the uncertain parameters affect the coefficients of the decision variables in the linear inequalities defining the safety sets. The interaction between the uncertain parameters and the variable coefficients in the safety set definition causes challenges in strengthening the original big-M formulations. By exploiting the connection between nominal chance-constrained programs and DR-CCP, we obtain strong formulations with significant enhancements. In particular, through this connection, we derive a linear number of valid inequalities, which can be immediately added to the formulations to obtain improved formulations in the original space of variables. In addition, we suggest a quantile-based strengthening procedure that allows us to reduce the big-M coefficients drastically. Furthermore, based on this procedure, we propose an exponential class of inequalities that can be separated efficiently within a branch-and-cut framework. The quantile-based strengthening procedure can be expensive. Therefore, for the special case of covering and packing type problems, we identify an efficient scheme to carry out this procedure. We demonstrate the computational efficacy of our proposed formulations on two classes of problems, namely stochastic portfolio optimization and resource planning.","PeriodicalId":73382,"journal":{"name":"INFORMS journal on optimization","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48928873","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Sparse principal component analysis and sparse canonical correlation analysis are two essential techniques from high-dimensional statistics and machine learning for analyzing large-scale data. Both problems can be formulated as an optimization problem with nonsmooth objective and nonconvex constraints. Because nonsmoothness and nonconvexity bring numerical difficulties, most algorithms suggested in the literature either solve some relaxations of them or are heuristic and lack convergence guarantees. In this paper, we propose a new alternating manifold proximal gradient method to solve these two high-dimensional problems and provide a unified convergence analysis. Numerical experimental results are reported to demonstrate the advantages of our algorithm.
{"title":"An Alternating Manifold Proximal Gradient Method for Sparse Principal Component Analysis and Sparse Canonical Correlation Analysis","authors":"Shixiang Chen, Shiqian Ma, Lingzhou Xue, H. Zou","doi":"10.1287/ijoo.2019.0032","DOIUrl":"https://doi.org/10.1287/ijoo.2019.0032","url":null,"abstract":"Sparse principal component analysis and sparse canonical correlation analysis are two essential techniques from high-dimensional statistics and machine learning for analyzing large-scale data. Both problems can be formulated as an optimization problem with nonsmooth objective and nonconvex constraints. Because nonsmoothness and nonconvexity bring numerical difficulties, most algorithms suggested in the literature either solve some relaxations of them or are heuristic and lack convergence guarantees. In this paper, we propose a new alternating manifold proximal gradient method to solve these two high-dimensional problems and provide a unified convergence analysis. Numerical experimental results are reported to demonstrate the advantages of our algorithm.","PeriodicalId":73382,"journal":{"name":"INFORMS journal on optimization","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1287/ijoo.2019.0032","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42788233","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This project spawned from an admission control problem we were working on for a major hospital in the Boston area. We tried to incorporate various aspects of the problem in a model, which resulted in a complex optimization problem that was difficult to solve analytically. Although numerical solutions could be computed, we were looking for insights to characterize simple policies that could be used in practice. We then came up with the idea of using machine learning to analyze solutions as a mean for obtaining such insights, an idea we thought could be interesting by itself. The motivating problem is an ongoing separate work.
{"title":"Mining Optimal Policies: A Pattern Recognition Approach to Model Analysis","authors":"Fernanda Bravo, Yaron Shaposhnik","doi":"10.2139/SSRN.3069690","DOIUrl":"https://doi.org/10.2139/SSRN.3069690","url":null,"abstract":"This project spawned from an admission control problem we were working on for a major hospital in the Boston area. We tried to incorporate various aspects of the problem in a model, which resulted in a complex optimization problem that was difficult to solve analytically. Although numerical solutions could be computed, we were looking for insights to characterize simple policies that could be used in practice. We then came up with the idea of using machine learning to analyze solutions as a mean for obtaining such insights, an idea we thought could be interesting by itself. The motivating problem is an ongoing separate work.","PeriodicalId":73382,"journal":{"name":"INFORMS journal on optimization","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49523262","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Gradient sampling (GS) methods for the minimization of objective functions that may be nonconvex and/or nonsmooth are proposed, analyzed, and tested. One of the most computationally expensive components of contemporary GS methods is the need to solve a convex quadratic subproblem in each iteration. By contrast, the methods proposed in this paper allow the use of inexact solutions of these subproblems, which, as proved in the paper, can be incorporated without the loss of theoretical convergence guarantees. Numerical experiments show that, by exploiting inexact subproblem solutions, one can consistently reduce the computational effort required by a GS method. Additionally, a strategy is proposed for aggregating gradient information after a subproblem is solved (potentially inexactly) as has been exploited in bundle methods for nonsmooth optimization. It is proved that the aggregation scheme can be introduced without the loss of theoretical convergence guarantees. Numerical experiments show that incorporating this gradient aggregation approach can also reduce the computational effort required by a GS method.
{"title":"Gradient Sampling Methods with Inexact Subproblem Solutions and Gradient Aggregation","authors":"Frank E. Curtis, Minhan Li","doi":"10.1287/ijoo.2022.0073","DOIUrl":"https://doi.org/10.1287/ijoo.2022.0073","url":null,"abstract":"Gradient sampling (GS) methods for the minimization of objective functions that may be nonconvex and/or nonsmooth are proposed, analyzed, and tested. One of the most computationally expensive components of contemporary GS methods is the need to solve a convex quadratic subproblem in each iteration. By contrast, the methods proposed in this paper allow the use of inexact solutions of these subproblems, which, as proved in the paper, can be incorporated without the loss of theoretical convergence guarantees. Numerical experiments show that, by exploiting inexact subproblem solutions, one can consistently reduce the computational effort required by a GS method. Additionally, a strategy is proposed for aggregating gradient information after a subproblem is solved (potentially inexactly) as has been exploited in bundle methods for nonsmooth optimization. It is proved that the aggregation scheme can be introduced without the loss of theoretical convergence guarantees. Numerical experiments show that incorporating this gradient aggregation approach can also reduce the computational effort required by a GS method.","PeriodicalId":73382,"journal":{"name":"INFORMS journal on optimization","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44283436","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We present a compact mixed integer program (MIP) for the backhaul profit maximization problem in which a freight carrier seeks to generate profit from an empty delivery vehicle’s backhaul trip from its last scheduled delivery to its depot by allowing it to deviate from the least expensive (or fastest) route to accept pickup-and-delivery requests between various points on the route as allowed by its capacity and required return time. The MIP is inspired by a novel representation of multicommodity flow that significantly reduces the size of the constraint matrix compared with a formulation based on the classical node-arc representation. This, in turn, leads to faster solution times when using a state-of-the-art MIP solver. In an empirical study of both formulations, problem instances with 10 potential pickup/drop-off locations and up to 72 pickup-and-delivery requests were solved an average 1.44 times faster in real time with our formulation, whereas instances with 20 locations and up to 332 pickup-and-delivery requests were solved an average of 11.88 times faster. The largest instances in the comparative study had 60 locations and up to 3,267 pickup-and-delivery requests; these instances required an average of more than 54 hours of real time to solve with the node-arc–based formulation but were solved in an average of under two hours of real time using our compact formulation. We also present a heuristic algorithm based on our compact formulation that finds near optimal solutions to each of the 60-location instances within 22 minutes of real time and near optimal solutions to instances with up to 80 locations within four and a half hours of real time.
{"title":"The Backhaul Profit Maximization Problem: Optimization Models and Solution Procedures","authors":"Yuanyuan Dong, Yulan Bai, E. Olinick, A. J. Yu","doi":"10.1287/ijoo.2022.0071","DOIUrl":"https://doi.org/10.1287/ijoo.2022.0071","url":null,"abstract":"We present a compact mixed integer program (MIP) for the backhaul profit maximization problem in which a freight carrier seeks to generate profit from an empty delivery vehicle’s backhaul trip from its last scheduled delivery to its depot by allowing it to deviate from the least expensive (or fastest) route to accept pickup-and-delivery requests between various points on the route as allowed by its capacity and required return time. The MIP is inspired by a novel representation of multicommodity flow that significantly reduces the size of the constraint matrix compared with a formulation based on the classical node-arc representation. This, in turn, leads to faster solution times when using a state-of-the-art MIP solver. In an empirical study of both formulations, problem instances with 10 potential pickup/drop-off locations and up to 72 pickup-and-delivery requests were solved an average 1.44 times faster in real time with our formulation, whereas instances with 20 locations and up to 332 pickup-and-delivery requests were solved an average of 11.88 times faster. The largest instances in the comparative study had 60 locations and up to 3,267 pickup-and-delivery requests; these instances required an average of more than 54 hours of real time to solve with the node-arc–based formulation but were solved in an average of under two hours of real time using our compact formulation. We also present a heuristic algorithm based on our compact formulation that finds near optimal solutions to each of the 60-location instances within 22 minutes of real time and near optimal solutions to instances with up to 80 locations within four and a half hours of real time.","PeriodicalId":73382,"journal":{"name":"INFORMS journal on optimization","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43126578","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Introduction to the Issue","authors":"D. Bertsimas","doi":"10.1287/ijoo.2019.0028","DOIUrl":"https://doi.org/10.1287/ijoo.2019.0028","url":null,"abstract":"","PeriodicalId":73382,"journal":{"name":"INFORMS journal on optimization","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1287/ijoo.2019.0028","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47731090","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We address the problem of accelerating column generation (CG) for set-covering formulations via dual optimal inequalities (DOIs). We study two novel classes of DOIs, which are referred to as Flexible DOIs (F-DOIs) and Smooth-DOIs (S-DOIs), respectively (and jointly as SF-DOIs). F-DOIs provide rebates for covering items more than necessary. S-DOIs describe the payment of a penalty to permit the undercoverage of items in exchange for the overinclusion of other items. Unlike other classes of DOIs from the literature, the S-DOIs and F-DOIs rely on very little problem-specific knowledge and, as such, have the potential to be applied to a vast number of problem domains. In particular, we discuss the application of the new DOIs to three relevant problems: the single-source capacitated facility location problem, the capacitated p-median problem, and the capacitated vehicle-routing problem. We provide computational evidence of the strength of the new inequalities by embedding them within a column-generation solver for these problems. Substantial speedups can be observed as when compared with a nonstabilized variant of the same CG procedure to achieve the linear-relaxation lower bound on problems with dense columns and structured assignment costs.
{"title":"Smooth and Flexible Dual Optimal Inequalities","authors":"Naveed Haghani, Claudio Contardo, Julian Yarkony","doi":"10.1287/ijoo.2021.0057","DOIUrl":"https://doi.org/10.1287/ijoo.2021.0057","url":null,"abstract":"We address the problem of accelerating column generation (CG) for set-covering formulations via dual optimal inequalities (DOIs). We study two novel classes of DOIs, which are referred to as Flexible DOIs (F-DOIs) and Smooth-DOIs (S-DOIs), respectively (and jointly as SF-DOIs). F-DOIs provide rebates for covering items more than necessary. S-DOIs describe the payment of a penalty to permit the undercoverage of items in exchange for the overinclusion of other items. Unlike other classes of DOIs from the literature, the S-DOIs and F-DOIs rely on very little problem-specific knowledge and, as such, have the potential to be applied to a vast number of problem domains. In particular, we discuss the application of the new DOIs to three relevant problems: the single-source capacitated facility location problem, the capacitated p-median problem, and the capacitated vehicle-routing problem. We provide computational evidence of the strength of the new inequalities by embedding them within a column-generation solver for these problems. Substantial speedups can be observed as when compared with a nonstabilized variant of the same CG procedure to achieve the linear-relaxation lower bound on problems with dense columns and structured assignment costs.","PeriodicalId":73382,"journal":{"name":"INFORMS journal on optimization","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42987731","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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