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小崎敏寛 (Toshihiro Kosaki)

線形計画問題に対する主双対内点法における 相補項の減少を考慮した 変数ごとのステップサイズの計算 Computation of Indexwise Step Sizes Considering Reduction of Complementarity Terms in a Primal-Dual Interior-Point Method for Linear Problems. 小崎敏寛 (Toshihiro Kosaki). Spectral Method( in Optimization) スペクトル法. Introduction General Framework

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小崎敏寛 (Toshihiro Kosaki)

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  1. 線形計画問題に対する主双対内点法における相補項の減少を考慮した変数ごとのステップサイズの計算Computation of Indexwise Step Sizes Considering Reduction of Complementarity Terms in a Primal-Dual Interior-Point Method for Linear Problems 小崎敏寛(Toshihiro Kosaki)

  2. Spectral Method( in Optimization)スペクトル法 • Introduction • General Framework • Interior-Point Method • Applications • Conclusions RIMS20122012/07/24

  3. 1 Introduction( Spectral Method) A (general) iterative algorithm for optimization problems x in R2 Δxk and are distinct. x* RIMS20122012/07/24

  4. 1 Spectral Method (cont.) • R6 • Spectral Method(スペクトル法) α:step size Δx:direction RIMS20122012/07/24

  5. 2 General Framework • An initial point is given. • If termination condition is satisfied then stop. • Compute a direction . • Compute step sizes . • Update . • Update k:=k+1. • Go to 2. RIMS20122012/07/24

  6. 2 General Framework (cont.) Features • An iterative algorithm with computing indexwise step sizes ‘s. • x is in a finite dimension. • x is a continuous variable. • Can compute easily all appropriate indexwise step sizes ‘s. (time of computing ) >>(time of computing ) RIMS20122012/07/24

  7. 3 (Primal-Dual) Interior-Point Method (for LP) • From an initial positive point w0:=(x0,y0,z0) with x0 >0 and z0 >0, keeping positivity and using both primal and dual problems, the algorithm seeks a point satisfying optimality condition. RIMS20122012/07/24

  8. 3 Interior-Point Method (cont.) x i z i↓0, i=1,…,n (optimality) ||r||↓0, (infeasibility) s.t. x>0 and z>0, where r is residual . RIMS20122012/07/24

  9. 3 Interior-Point Method (cont.) central path Δw Most of computational cost is solving normal equations: RIMS20122012/07/24

  10. 3 Interior-Point Method (cont.) • (total computational time) =(number of iterations)×(time per iteration) Predictor-corrector method Multiple-centering method CG method RIMS20122012/07/24

  11. 3 Proposition1 Proposition1:Relation between positivity and xizi xi(α):=xi(0)+αΔxi, zi(α):=zi(0)+αΔzi (i=1,…,n). Suppose xi(0)>0 and zi(0)>0 (i=1,…,n). RIMS20122012/07/24

  12. 3 Proposition2 Proposition2:Relation between step size α and xTz xTz α2Δ xTΔz γxTz α 1 0 Neglecting quadratic term, reduction of xTz is in proportion to α. RIMS20122012/07/24

  13. 3 Proposition3 Proposition3:Relation between step size αand residual r r1 r2 α 1 Reduction of r’s is in proportion to α. RIMS20122012/07/24

  14. 3 Interior-Point Method (cont.) • IPM has been extended from LP to QP, LCP, SOCP and SDP. RIMS20122012/07/24

  15. 4 Applications • LP1 • LP2 • SDP • SOCP RIMS20122012/07/24

  16. 4-1 LP1 e.g. αc=0.99995 Computation of Δy Computation of αi* ’s Cholesky decomposition ADAT O(m3) O(n) O(m2n) RIMS20122012/07/24

  17. 4-1 LP1 (cont.) • Numerical experiment • Transportation problem • M: # of supply nodes, N: # of demand nodes • LP with M×N variables RIMS20122012/07/24

  18. 4-2 LP2 • Minimizing →Analytical solutions are availablefrom quadratic equations xi(αi):=xi(0)+α i Δxi, zi(αi):=zi(0)+αiΔzi (i=1,…,n). • The smaller xi zi is, the better it is. • The larger αi is, the better it is. x i z i ideal α i 0 1 RIMS20122012/07/24

  19. 4-3 SDP Block diagonal SDP RIMS20122012/07/24

  20. 4-4 SOCP Direct product of second order cones variables: x1 x2 Δx1 × Δx2 K2 K1 RIMS20122012/07/24

  21. 5 Conclusions • We proposed a general framework (Spectral Method). • The framework was applied to some optimization problems. RIMS20122012/07/24

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