The Best and the Worst: Computing the Optimal Value Range in - PowerPoint PPT Presentation

The Best and the Worst: Computing the Optimal Value Range in Interval Linear Programming 1 Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University 2 Faculty of Finance and Accounting, University of Economics, Prague The 17th International Conference on Operational Research (KOI 2018), Zadar Elif Garajová 1 , Milan Hladík 1 & Miroslav Rada 2

approximation and rounding estimating the future representing missing data inexact measurements discretization of time 22 °C 23 5 °C 3 14159 0 05 0 66 0 21 0 84 d t max t min Interval Linear Programming b 0 05 g 5 a €17 1 c €15 6 1 Consider a linear programming problem . . . minimize c T x subject to Ax ≤ b

Interval Linear Programming 1 approximation and rounding b ≈ 3 . 14159 Consider a linear programming problem . . . minimize c T x subject to Ax ≤ b estimating the future €15 . 6 ≤ c ≤ €17 . 1 representing missing data 0 . 66 , 0 . 21 , 0 . 84 , d =? , 0 . 05 inexact measurements discretization of time a = 5 ± 0 . 05 g t min = 22 °C , t max = 23 . 5 °C

Interval Linear Programming 1 approximation and rounding [ b ] = [ 3 . 141592 , 3 . 141593 ] Consider an interval linear programming problem . . . minimize [ c ] T x subject to [ A ] x ≤ [ b ] estimating the future representing missing data [ c ] = [ 15 . 6 , 17 . 1 ] [ d ] = [ 0 , 1 ] inexact measurements discretization of time [ a ] = [ 4 . 95 , 5 . 05 ] [ t ] = [ 22 , 23 . 5 ]

Interval Linear Programming: Definitions interval matrix as the set • An interval linear program is a family of linear programs minimize subject to 2 • Given two real matrices A , A ∈ R m × n with A ≤ A , we define an [ A ] = [ A , A ] = { A ∈ R m × n : A ≤ A ≤ A } . a T x + c T y Ax + By = b , Cx + Dy ≤ d , x ≥ 0 , where A ∈ [ A ] , B ∈ [ B ] , C ∈ [ C ] , D ∈ [ D ] , a ∈ [ a ] , b ∈ [ b ] , c ∈ [ c ] , d ∈ [ d ] . • A linear program in the family is called a scenario .

Interval Linear Programming: Definitions minimize • A linear program in the family is called a scenario . subject to 2 interval matrix as the set • Given two real matrices A , A ∈ R m × n with A ≤ A , we define an [ A ] = [ A , A ] = { A ∈ R m × n : A ≤ A ≤ A } . • An interval linear program is a family of linear programs a T x + c T y [ a ] T x = b is not Ax + By = b , [ a ] T x ≤ b , [ a ] T x ≥ b Cx + Dy ≤ d , x ≥ 0 , where A ∈ [ A ] , B ∈ [ B ] , C ∈ [ C ] , D ∈ [ D ] , a ∈ [ a ] , b ∈ [ b ] , c ∈ [ c ] , d ∈ [ d ] .

Interval Linear Programming: Definitions interval matrix as the set minimize c T x subject to • A linear program in the family is called a scenario . 2 • Given two real matrices A , A ∈ R m × n with A ≤ A , we define an [ A ] = [ A , A ] = { A ∈ R m × n : A ≤ A ≤ A } . • An interval linear program is a family of linear programs Ax = b , x ≥ 0 , where A ∈ [ A ] , b ∈ [ b ] , c ∈ [ c ] .

Interval Linear Programming: Definitions interval matrix as the set minimize c T x subject to • A linear program in the family is called a scenario . 2 • Given two real matrices A , A ∈ R m × n with A ≤ A , we define an [ A ] = [ A , A ] = { A ∈ R m × n : A ≤ A ≤ A } . • An interval linear program is a family of linear programs Ax = b , x ≥ 0 , where A ∈ [ A ] , b ∈ [ b ] , c ∈ [ c ] .

Feasibility and Optimality • A vector x is a weakly feasible/optimal solution to the interval program, if x is a feasible/optimal solution for some scenario. • A vector x is a strongly feasible/optimal solution to the interval program, if x is a feasible/optimal solution for each scenario. the worst optimal value (or the optimal value range ) 3 • Regarding optimal values, we usually consider the best and f ([ A ] , [ b ] , [ c ]) = inf { f ( A , b , c ) : A ∈ [ A ] , b ∈ [ b ] , c ∈ [ c ] } , f ([ A ] , [ B ] , [ c ]) = sup { f ( A , b , c ) : A ∈ [ A ] , b ∈ [ b ] , c ∈ [ c ] } .

Feasibility and Optimality • A vector x is a weakly feasible/optimal solution to the interval program, if x is a feasible/optimal solution for some scenario. • A vector x is a strongly feasible/optimal solution to the interval program, if x is a feasible/optimal solution for each scenario. • Regarding optimal values, we usually consider the best and the worst optimal value (or the optimal value range ) 3 f ([ A ] , [ b ] , [ c ]) = inf { f ( A , b , c ) : A ∈ [ A ] , b ∈ [ b ] , c ∈ [ c ] } , f ([ A ] , [ B ] , [ c ]) = sup { f ( A , b , c ) : A ∈ [ A ] , b ∈ [ b ] , c ∈ [ c ] } .

Feasibility and Optimality • A vector x is a weakly feasible/optimal solution to the interval program, if x is a feasible/optimal solution for some scenario. • A vector x is a strongly feasible/optimal solution to the interval program, if x is a feasible/optimal solution for each scenario. the worst optimal value (or the optimal value range ) 3 • Regarding optimal values, we usually consider the best and f ([ A ] , [ b ] , [ c ]) = inf { f ( A , b , c ) : A ∈ [ A ] , b ∈ [ b ] , c ∈ [ c ] } , f ([ A ] , [ B ] , [ c ]) = sup { f ( A , b , c ) : A ∈ [ A ] , b ∈ [ b ] , c ∈ [ c ] } .

Interval Linear Programming: Example maximize x 2 subject to 4 [ − 1 , 1 ] x 1 + x 2 ≤ 0 x 2 ≤ 1

Interval Linear Programming: Example 2 1 x 1 x 2 0 1 -1 4 maximize 3 1 -1 -2 -3 -4 subject to x 2 4 Let's traverse through this! [ − 1 , 1 ] x 1 + x 2 ≤ 0 x 2 ≤ 1 − 1 − 1

Interval Linear Programming: Example 2 1 x 1 x 2 0 1 -1 4 maximize 3 1 -1 -2 -3 -4 subject to x 2 4 Let's traverse through this! [ − 1 , 1 ] x 1 + x 2 ≤ 0 x 2 ≤ 1 − 0 . 5 − 1

Interval Linear Programming: Example 2 1 x 1 x 2 0 1 -1 4 maximize 3 1 -1 -2 -3 -4 subject to x 2 4 Let's traverse through this! [ − 1 , 1 ] x 1 + x 2 ≤ 0 x 2 ≤ 1 − 0 . 33 − 1

Interval Linear Programming: Example 2 1 x 1 x 2 0 1 -1 4 maximize 3 1 -1 -2 -3 -4 subject to x 2 4 Let's traverse through this! [ − 1 , 1 ] x 1 + x 2 ≤ 0 x 2 ≤ 1 − 0 . 25 − 1

Interval Linear Programming: Example maximize 0 1 x 1 x 2 0 1 -1 4 3 2 1 -1 -2 -3 -4 subject to x 2 4 Let's traverse through this! [ − 1 , 1 ] x 1 + x 2 ≤ 0 x 2 ≤ 1 − 1

Interval Linear Programming: Example 2 1 x 1 x 2 0 1 -1 4 maximize 3 1 -1 -2 -3 -4 subject to x 2 4 Let's traverse through this! [ − 1 , 1 ] x 1 + x 2 ≤ 0 x 2 ≤ 1 0 . 25 − 1

Interval Linear Programming: Example 2 1 x 1 x 2 0 1 -1 4 maximize 3 1 -1 -2 -3 -4 subject to x 2 4 Let's traverse through this! [ − 1 , 1 ] x 1 + x 2 ≤ 0 x 2 ≤ 1 0 . 33 − 1

Interval Linear Programming: Example 2 1 x 1 x 2 0 1 -1 4 maximize 3 1 -1 -2 -3 -4 subject to x 2 4 Let's traverse through this! [ − 1 , 1 ] x 1 + x 2 ≤ 0 x 2 ≤ 1 0 . 5 − 1

Interval Linear Programming: Example maximize 1 1 x 1 x 2 0 1 -1 4 3 2 1 -1 -2 -3 -4 subject to x 2 4 Let's traverse through this! [ − 1 , 1 ] x 1 + x 2 ≤ 0 x 2 ≤ 1 − 1

Interval Linear Programming: Example 2 x 1 x 2 0 1 -1 4 3 1 maximize -1 -2 -3 -4 subject to x 2 4 [ − 1 , 1 ] x 1 + x 2 ≤ 0 x 2 ≤ 1 Optimal values: { 0 , 1 }

Computing the Optimal Value Range Best optimal value: Worst optimal value: Since computing the worst optimal value f exactly is difficult, we can try to find an upper bound f L (e.g. iterative improvement from a scenario or relaxations). 5 f = inf c T x : Ax ≤ b , Ax ≥ b , x ≥ 0 Theorem (Oettli, Prager, 1964): x solves [ A ] x = [ b ] ⇔ | A c x − b c | ≤ A ∆ | x | + b ∆ f = sup s ∈{± 1 } m f ( A c − diag ( s ) A ∆ , b c + diag ( s ) b ∆ , c ) Theorem (Rohn, 1997): Deciding whether f ( A , [ b ] , c ) ≥ 1 holds is NP-hard for interval linear programs of type min c T x : Ax = [ b ] , x ≥ 0 . U and a lower bound f

• … Semi-strong Optimality 1 Luo, J., Li, W., Strong optimal solutions of interval linear programming (2013). 6 A vector x ∈ R n is 1 … • a ( ∅ ) -strong optimal solution of the ILP if it is an optimal solution for some scenario with A ∈ [ A ] , b ∈ [ b ] , c ∈ [ c ] . • a ([ c ]) -strong optimal solution of the ILP if for each c ∈ [ c ] there exist A ∈ [ A ] , b ∈ [ b ] such that x is optimal for the scenario ( A , b , c ) . • a ([ b ]) -strong optimal solution of the ILP if for each b ∈ [ b ] there exist A ∈ [ A ] , c ∈ [ c ] such that x is optimal for the scenario ( A , b , c ) . • a ([ b ] , [ c ]) -strong optimal solution of the ILP if for each b ∈ [ b ] , c ∈ [ c ] there exists A ∈ [ A ] such that x is optimal for the scenario ( A , b , c ) . • an ([ A ] , [ b ] , [ c ]) -strong optimal solution of the ILP if it is an optimal solution for each scenario with A ∈ [ A ] , b ∈ [ b ] , c ∈ [ c ] .

From Optimal Values to Semi-strong Values Let us now reformulate the problem of computing the optimal value range… each scenario. Then, the best and the worst optimal value can be viewed as the best of all weak or strong values, respectively. 7 • A value r ∈ R is a weak value , if there is a scenario of the program with f ( A , b , c ) ≤ r . • A value r ∈ R is a strong value , if f ( A , b , c ) ≤ r holds for