tailieunhanh - David G. Luenberger, Yinyu Ye - Linear and Nonlinear Programming International Series Episode 2 Part 11

Tham khảo tài liệu 'david g. luenberger, yinyu ye - linear and nonlinear programming international series episode 2 part 11', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Semidefinite Programming 497 transformed to semidefinite constraints and hence the entire problem converted to a semidefinite program. This approach is useful in many applications especially in various problems of control theory. As in other instances of duality the duality of semidefinite programs is weak unless other conditions hold. We state here but do not prove a version of the strong duality theorem. Strong Duality in SDP. Suppose SDP and SDD are both feasible and at least one of them has an interior. Then there are optimal solutions to the primal and the dual and their optimal values are equal. If the non-empty interior condition of the above theorem does not hold then the duality gap may not be zero at optimality. Example 8 The following semidefinite program has a duality gap 0 1 0 0 0 0 0 -1 0 C 1 0 0 A1 0 1 0 A2 -1 0 0 0 0 0 0 0 0 0 0 2_ and b 0 10 The primal minimal objective value is 0 achieved by 0 0 0 X 0 0 0 0 0 5 and the dual maximal objective value is -10 achieved by y 0 -1 so the duality gap is 10. Interior-Point Algorithms for SDP Let the primal SDP and dual SDD semidefinite programs both have interior point feasible solutions. Then the central path can be expressed as e Ị X y S e XS I 0 p . The primal-dual potential function for SDP a descent merit function is K p X S n p log X S - log det X det S where p 0. Note that if X and S are diagonal matrices these definitions reduce to those for linear programming. 498 Chapter 15 Primal-Dual Methods Once we have an interior feasible point X y S we can generate a new iterate X y S by solving for DX dy DS from the primal-dual system of linear equations D-1DXD-1 DS 772 X - S A DX 0 for all i 64 - Em dy A - Ds 0 where D is the scaling matrix __ Vĩ A1 CTỈ1- 2 Y 2 X 2 X 2 SX 2 I 2 X 2 and X S n. Then one assigns X X ãDX y y ãdy and S s ãDS where ã arg min X DX S DS . a 0 n p Furthermore it can be shown that ftn p X S - ỳ P X S -Ỗ for a constant Ỗ . This provides an iteration complexity bound that is .