A tight characterization of the performance of static solutions in two-stage adjustable robust linear optimization
In this paper, we study the performance of static solutions for two-stage adjustable robust linear optimization problems with uncertain constraint and objective coefficients and give a tight characterization of the adaptivity gap. Computing an optimal adjustable robust optimization problem is often...
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description | In this paper, we study the performance of static solutions for two-stage adjustable robust linear optimization problems with uncertain constraint and objective coefficients and give a tight characterization of the adaptivity gap. Computing an optimal adjustable robust optimization problem is often intractable since it requires to compute a solution for all possible realizations of uncertain parameters (Feige et al. in Lect Notes Comput Sci 4513:439–453,
2007
). On the other hand, a static solution is a single (here and now) solution that is feasible for all possible realizations of the uncertain parameters and can be computed efficiently for most dynamic optimization problems. We show that for a fairly general class of uncertainty sets, a static solution is optimal for the two-stage adjustable robust linear packing problems. This is highly surprising in view of the usual perception about the conservativeness of static solutions. Furthermore, when a static solution is not optimal for the adjustable robust problem, we give a tight approximation bound on the performance of the static solution that is related to a measure of non-convexity of a transformation of the uncertainty set. We also show that our bound is at least as good (and in many case significantly better) as the bound given by the symmetry of the uncertainty set (Bertsimas and Goyal in Math Methods Oper Res 77(3):323–343,
2013
; Bertsimas et al. in Math Oper Res 36(1):24–54,
2011
). |
doi_str_mv | 10.1007/s10107-014-0768-y |
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2007
). On the other hand, a static solution is a single (here and now) solution that is feasible for all possible realizations of the uncertain parameters and can be computed efficiently for most dynamic optimization problems. We show that for a fairly general class of uncertainty sets, a static solution is optimal for the two-stage adjustable robust linear packing problems. This is highly surprising in view of the usual perception about the conservativeness of static solutions. Furthermore, when a static solution is not optimal for the adjustable robust problem, we give a tight approximation bound on the performance of the static solution that is related to a measure of non-convexity of a transformation of the uncertainty set. We also show that our bound is at least as good (and in many case significantly better) as the bound given by the symmetry of the uncertainty set (Bertsimas and Goyal in Math Methods Oper Res 77(3):323–343,
2013
; Bertsimas et al. in Math Oper Res 36(1):24–54,
2011
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2007
). On the other hand, a static solution is a single (here and now) solution that is feasible for all possible realizations of the uncertain parameters and can be computed efficiently for most dynamic optimization problems. We show that for a fairly general class of uncertainty sets, a static solution is optimal for the two-stage adjustable robust linear packing problems. This is highly surprising in view of the usual perception about the conservativeness of static solutions. Furthermore, when a static solution is not optimal for the adjustable robust problem, we give a tight approximation bound on the performance of the static solution that is related to a measure of non-convexity of a transformation of the uncertainty set. We also show that our bound is at least as good (and in many case significantly better) as the bound given by the symmetry of the uncertainty set (Bertsimas and Goyal in Math Methods Oper Res 77(3):323–343,
2013
; Bertsimas et al. in Math Oper Res 36(1):24–54,
2011
).</description><subject>Adjustable</subject><subject>Approximation</subject><subject>Calculus of Variations and Optimal Control; Optimization</subject><subject>Combinatorics</subject><subject>Computation</subject><subject>Decision making</subject><subject>Expected values</subject><subject>Full Length Paper</subject><subject>Mathematical analysis</subject><subject>Mathematical and Computational Physics</subject><subject>Mathematical Methods in Physics</subject><subject>Mathematical models</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Mathematics of Computing</subject><subject>Numerical Analysis</subject><subject>Operations research</subject><subject>Optimization</subject><subject>Perception</subject><subject>Revenue management</subject><subject>Studies</subject><subject>Theoretical</subject><subject>Transformations</subject><subject>Uncertainty</subject><issn>0025-5610</issn><issn>1436-4646</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2015</creationdate><recordtype>article</recordtype><sourceid>ABUWG</sourceid><sourceid>AFKRA</sourceid><sourceid>AZQEC</sourceid><sourceid>BENPR</sourceid><sourceid>CCPQU</sourceid><sourceid>DWQXO</sourceid><sourceid>GNUQQ</sourceid><recordid>eNp1kU9LxDAQxYMouK5-AG8BL16ikyZN2uOy-A8EL3oOaZrudmmbNUmR9dOb0j2I4GmGmd97zPAQuqZwRwHkfaBAQRKgnIAUBTmcoAXlTBAuuDhFC4AsJ7mgcI4uQtgBAGVFsUB-hWO72UZsttprE61vv3Vs3YBdg-PW4r31jfO9HoydRiGmrcHBdeNEBdwOOH45kuYbi3W9G1NXdRZ7V6UWd-1gtcduH9v-6HyJzhrdBXt1rEv08fjwvn4mr29PL-vVKzFcZJEIyjNbMlHWeVULXmhpcsk0ZIVpqGEyT__WTDYFq4A1sqpqk0NVW81kBSW3bIluZ9-9d5-jDVH1bTC26_Rg3RgUFVKWlALLE3rzB9250Q_pukQJziHLCpooOlPGuxC8bdTet732B0VBTSmoOQWVUlBTCuqQNNmsCYkdNtb_cv5X9APow4xz</recordid><startdate>20150501</startdate><enddate>20150501</enddate><creator>Bertsimas, Dimitris</creator><creator>Goyal, Vineet</creator><creator>Lu, Brian Y.</creator><general>Springer Berlin Heidelberg</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><scope>3V.</scope><scope>7SC</scope><scope>7WY</scope><scope>7WZ</scope><scope>7XB</scope><scope>87Z</scope><scope>88I</scope><scope>8AL</scope><scope>8AO</scope><scope>8FD</scope><scope>8FE</scope><scope>8FG</scope><scope>8FK</scope><scope>8FL</scope><scope>ABJCF</scope><scope>ABUWG</scope><scope>AFKRA</scope><scope>ARAPS</scope><scope>AZQEC</scope><scope>BENPR</scope><scope>BEZIV</scope><scope>BGLVJ</scope><scope>CCPQU</scope><scope>DWQXO</scope><scope>FRNLG</scope><scope>F~G</scope><scope>GNUQQ</scope><scope>HCIFZ</scope><scope>JQ2</scope><scope>K60</scope><scope>K6~</scope><scope>K7-</scope><scope>L.-</scope><scope>L6V</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope><scope>M0C</scope><scope>M0N</scope><scope>M2P</scope><scope>M7S</scope><scope>P5Z</scope><scope>P62</scope><scope>PQBIZ</scope><scope>PQBZA</scope><scope>PQEST</scope><scope>PQQKQ</scope><scope>PQUKI</scope><scope>PTHSS</scope><scope>Q9U</scope></search><sort><creationdate>20150501</creationdate><title>A tight characterization of the performance of static solutions in two-stage adjustable robust linear optimization</title><author>Bertsimas, Dimitris ; Goyal, Vineet ; Lu, Brian Y.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c462t-6142e9369d5bd648a7c573a028cf1c375107d37f83b03f7bbdc50bdea37b094e3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2015</creationdate><topic>Adjustable</topic><topic>Approximation</topic><topic>Calculus of Variations and Optimal Control; Optimization</topic><topic>Combinatorics</topic><topic>Computation</topic><topic>Decision making</topic><topic>Expected values</topic><topic>Full Length Paper</topic><topic>Mathematical analysis</topic><topic>Mathematical and Computational Physics</topic><topic>Mathematical Methods in Physics</topic><topic>Mathematical models</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Mathematics of Computing</topic><topic>Numerical Analysis</topic><topic>Operations research</topic><topic>Optimization</topic><topic>Perception</topic><topic>Revenue management</topic><topic>Studies</topic><topic>Theoretical</topic><topic>Transformations</topic><topic>Uncertainty</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Bertsimas, Dimitris</creatorcontrib><creatorcontrib>Goyal, Vineet</creatorcontrib><creatorcontrib>Lu, Brian Y.</creatorcontrib><collection>CrossRef</collection><collection>ProQuest Central (Corporate)</collection><collection>Computer and Information Systems Abstracts</collection><collection>Access via ABI/INFORM (ProQuest)</collection><collection>ABI/INFORM Global (PDF only)</collection><collection>ProQuest Central (purchase pre-March 2016)</collection><collection>ABI/INFORM Global (Alumni Edition)</collection><collection>Science Database (Alumni Edition)</collection><collection>Computing Database (Alumni Edition)</collection><collection>ProQuest Pharma Collection</collection><collection>Technology Research Database</collection><collection>ProQuest SciTech Collection</collection><collection>ProQuest Technology Collection</collection><collection>ProQuest Central (Alumni) (purchase pre-March 2016)</collection><collection>ABI/INFORM Collection (Alumni Edition)</collection><collection>Materials Science & Engineering Collection</collection><collection>ProQuest Central (Alumni Edition)</collection><collection>ProQuest Central UK/Ireland</collection><collection>Advanced Technologies & Aerospace Collection</collection><collection>ProQuest Central Essentials</collection><collection>ProQuest Central</collection><collection>Business Premium Collection</collection><collection>Technology Collection</collection><collection>ProQuest One Community College</collection><collection>ProQuest Central Korea</collection><collection>Business Premium Collection (Alumni)</collection><collection>ABI/INFORM Global (Corporate)</collection><collection>ProQuest Central Student</collection><collection>SciTech Premium Collection</collection><collection>ProQuest Computer Science Collection</collection><collection>ProQuest Business Collection (Alumni Edition)</collection><collection>ProQuest Business Collection</collection><collection>Computer Science Database</collection><collection>ABI/INFORM Professional Advanced</collection><collection>ProQuest Engineering Collection</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>Computer and Information Systems Abstracts Academic</collection><collection>Computer and Information Systems Abstracts Professional</collection><collection>ABI/INFORM Global</collection><collection>Computing Database</collection><collection>Science Database</collection><collection>Engineering Database</collection><collection>Advanced Technologies & Aerospace Database</collection><collection>ProQuest Advanced Technologies & Aerospace Collection</collection><collection>ProQuest One Business</collection><collection>ProQuest One Business (Alumni)</collection><collection>ProQuest One Academic Eastern Edition (DO NOT USE)</collection><collection>ProQuest One Academic</collection><collection>ProQuest One Academic UKI Edition</collection><collection>Engineering Collection</collection><collection>ProQuest Central Basic</collection><jtitle>Mathematical programming</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Bertsimas, Dimitris</au><au>Goyal, Vineet</au><au>Lu, Brian Y.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>A tight characterization of the performance of static solutions in two-stage adjustable robust linear optimization</atitle><jtitle>Mathematical programming</jtitle><stitle>Math. Program</stitle><date>2015-05-01</date><risdate>2015</risdate><volume>150</volume><issue>2</issue><spage>281</spage><epage>319</epage><pages>281-319</pages><issn>0025-5610</issn><eissn>1436-4646</eissn><coden>MHPGA4</coden><abstract>In this paper, we study the performance of static solutions for two-stage adjustable robust linear optimization problems with uncertain constraint and objective coefficients and give a tight characterization of the adaptivity gap. Computing an optimal adjustable robust optimization problem is often intractable since it requires to compute a solution for all possible realizations of uncertain parameters (Feige et al. in Lect Notes Comput Sci 4513:439–453,
2007
). On the other hand, a static solution is a single (here and now) solution that is feasible for all possible realizations of the uncertain parameters and can be computed efficiently for most dynamic optimization problems. We show that for a fairly general class of uncertainty sets, a static solution is optimal for the two-stage adjustable robust linear packing problems. This is highly surprising in view of the usual perception about the conservativeness of static solutions. Furthermore, when a static solution is not optimal for the adjustable robust problem, we give a tight approximation bound on the performance of the static solution that is related to a measure of non-convexity of a transformation of the uncertainty set. We also show that our bound is at least as good (and in many case significantly better) as the bound given by the symmetry of the uncertainty set (Bertsimas and Goyal in Math Methods Oper Res 77(3):323–343,
2013
; Bertsimas et al. in Math Oper Res 36(1):24–54,
2011
).</abstract><cop>Berlin/Heidelberg</cop><pub>Springer Berlin Heidelberg</pub><doi>10.1007/s10107-014-0768-y</doi><tpages>39</tpages><oa>free_for_read</oa></addata></record> |
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subjects | Adjustable Approximation Calculus of Variations and Optimal Control Optimization Combinatorics Computation Decision making Expected values Full Length Paper Mathematical analysis Mathematical and Computational Physics Mathematical Methods in Physics Mathematical models Mathematics Mathematics and Statistics Mathematics of Computing Numerical Analysis Operations research Optimization Perception Revenue management Studies Theoretical Transformations Uncertainty |
title | A tight characterization of the performance of static solutions in two-stage adjustable robust linear optimization |
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