Linear Vector Optimization. Algorithms and Applications Andreas Löhne Martin-Luther-Universität Halle-Wittenberg, Germany ANZIAM 2013 Newcastle (Australia), February 4, 2013
based on Hamel, A.; Löhne, A.; Rudlo, B: A Benson type algorithm for linear vector optimization and applications, almost submittted
Problem Compute where P [S] + C P [S] := {P x x S}, P R q n S := {x R n Ax b}, A R m n, b R m C := { y R q Z T y 0 }, Z R q p Throughout we assume the cone C being pointed and solid.
Polyhedra P... convex polyhedron in R q H-representation... intersection of halfspaces: P = r i=1 { y R q (z i ) T y γ i } V-representation... generalized convex hull of generating points y 1,... y r R q and generating directions k 1,... k s R q \ {0}: P = conv (y 1,..., y s ) + cone (k 1,..., k t )
Problem Compute where P [S] + C P [S] := {P x x S}, P R q n S := {x R n Ax b}, A R m n, b R m C := { y R q Z T y 0 }, Z R q p
Special case: q=1 Compute where p T [S] + R + p T [S] := { p T x x S }, p R n S := {x R n Ax b}, A R m n, b R m Linear Program
Algorithm P [S]
Algorithm P := P [S] + C Notation: P := P [S] + C
Weighted sum scalarization (P 1 (w)) min w T P x s.t. Ax b (D 1 (w)) max b T u s.t. A T u = P T w, u 0 w... columns of Z C := { y R q Z T y 0 }
Algorithm P := P [S] + C
Algorithm T t
Translative scalarization (P 2 (y)) min z s.t. Ax b, Z T P x Z T y + z Z T c y R q, c int C
Translative scalarization (P 2 (y)) min z s.t. Ax b, Z T P x Z T y + z Z T c y R q, c int C
Translative scalarization (P 2 (y)) min z s.t. Ax b, Z T P x Z T y + z Z T c ( D 2 (y)) max b T u y T Zv subject to A T u P T Zv = 0 c T Zv = 1 (u, v) 0. (D 2 (y)) max b T u y T w subject to A T u P T w = 0 c T w = 1 Y T w 0 u 0, Y... matrix of generating vectors of C Z... matrix of generating vectors of C + Y T w 0 y C : y T w 0 w C + v 0 : w = Zv
Translative scalarization Proposition. Let S and c int C. For every t R q, there exist optimal solutions ( x, z) to (P 2 (t)) and (ū, w) to (D 2 (t)). Each solution (ū, w) to (D 2 (t)) denes a supporting hyperplane H := { y R q w T y = b T ū } of P := P [S] + C such that s := t + z c H P. We have t P z < 0, t bd P z = 0, t int P z > 0.
Algorithm T t
Algorithm T t H
Algorithm
Algorithm t H
Algorithm t 1 t 2
New variant of Benson's algorithm Input: Ha B, b, P, Z (data of (P)); Ha a solution ({0}, S h ) to (P h ); Ha a solution T h to (D h ); Output: Ha ( S, S h ) is a solution to (P); Ha T is a solution to (D ); Ha ( T p, ˆT p) is a V -representation of P; Ha ( T d, (0,..., 0, 1) T ) is a V -representation of D ;
Ha T {( solve(d 1 (w)), w ) (u, w) T h} ; Ha ag true; Ha while (ag) HaHa ag false; HaHa S ; HaHa T d { D (u, w) (u, w) T } ; HaHa (T p, ˆT p ) dual(t d, (0,..., 0, 1) T ); HaHa for i = 1 to T p do HaHaHa t T p [i]; HaHaHa (x, z, u, w) solve(p 2 (t)/d 2 (t)); HaHaHa if z > ε HaHaHaHa T T {(u, w)}; HaHaHaHa ag=true; HaHaHaHa break; (optional) HaHaHa else HaHaHaHa S S {x}; HaHaHa end; HaHa end; Ha end;
Advantages only one LP per iteration step (rather than two or three) LPs have (essentially) the same matrix (good for warm starts) fewer iteration steps to obtain ε-solution
Numerical examples Implementation with Matlab LP solver: GLPK Vertex enumeration: CDDLIB Graphics: Javaview (www.javaview.de) Constraints of type a Bx b, lb x ub
Radio therapie treatment planning [Shao & Ehrgott, 2008] matrix size: 1211 1143 (153 930 nonzeros) objectives: 3 ordering cone: R 3 + ε variant total time S T # LPs t max /t aver 0.3 break 47 secs 46 29 75 1.8 0.05 break 144 secs 176 94 270 2.0 0.005 break 1596 secs 1456 597 2053 1.9
Radio therapy treatment planning [Shao & Ehrgott, 2008] ε = 0.3 ε = 0.05
Specialized parametric simplex method [Ruszczy«ski&Vanderbei, 2003] matrix size: 6161 3799 (4 435 919 nonzero entries) objectives: 2 ordering cone: R 2 + ε total time S T # LPs t max t max /t aver 2 10 4 946 secs 6 7 13 347 secs 4.4 2 10 5 1648 secs 22 23 45 304 secs 8.9 2 10 6 3085 secs 62 63 125 310 secs 14.1 In R&V03: not much more time required to get an exact solution than for solving one single LP!!!
Specialized parametric simplex method [Ruszczy«ski&Vanderbei, 2003] ε = 2 10 4 ε = 2 10 6
Set-valued Average Value at Risk matrix size: 24 586 295 056 (1 150 986 nonzero entries) objectives: 2 ordering cone: C R 2 + ε total time S S h T # LPs t max t max /t aver 10 4 3529 secs 20 1 21 46 592 secs 8.4 10 5 4716 secs 47 1 48 100 671 secs 17.1 10 6 7905 secs 122 1 123 253 449 secs 22.0
Set-valued Average Value at Risk primal dual ε = 10 4
Set-valued Average Value at Risk 3 objectives 4 objectives cone has 6 extreme directions cone has 12 extreme directions 11 272 124 025 matrix 5 126 51 300 matrix 1748 secs for ε = 10 4 384 secs for ε = 10 2
Literature - History Benson, H. P.: An outer approximation algorithm for generating all ecient extreme points in the outcome set of a multiple objective linear programming problem. Journal of Global Optimization 13, (1998) Heyde, F., Löhne, A.: Geometric duality in multiple objective linear programming. SIAM Journal of Optimization 19 (2), (2008) Ehrgott, M.; Löhne, A.; Shao, L.: A dual variant of Benson's outer approximation algorithm. J. Glob. Opt. 52 (4), (2011) (submitted 2007) Shao, L. and Ehrgott, M.: Approximately solving multiobjective linear programmes in objective space and an application in radiotherapy treatment planning. Math. Methods Oper. Res. 68(2), (2008) Löhne, A.: Vector optimization with inmum and supremum. Springer (2011) (extension to unbounded problems) Hamel, A., Löhne, A., Rudlo, B: A Benson type algorithm for linear vector optimization and applications, almost submittted (arbitrary cones, one LP per step) Löhne, A., Schrage, C.: An algorithm to solve polyhedral convex set optimization problems, Optimization, 62(1), (2013)