By Henry Wolkowicz, Romesh Saigal, Lieven Vandenberghe
Advent / Henry Wolkowicz, Ramesh Saigal and Lieven Vandenberghe -- Pt. I. conception. Convex research on Symmetric Matrices / Florian Jarre. The Geometry of Semidefinite Programming / Gabor Pataki. Duality and Optimality stipulations / Alexander Shapiro and Katya Scheinberg. Self-Dual Embeddings / Etienne de Klerk, Tamas Terlaky and Kees Roos. Robustness / Aharon Ben-Tal, Laurent El Ghaoui and Arkadi Nemirovski. errors research / Zhiquan Luo and Jos Sturm -- Pt. II. Algorithms. Symmetric Cones, power aid equipment / Farid Alizadeh and Stefan Schmieta. capability relief and Primal-Dual tools / Levent Tuncel. Path-Following tools / Renato Monteiro and Michael Todd
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Extra info for Handbook of semidefinite programming : theory, algorithms, and applications
Example text
Xr = 1, (j) (j) (j) xq+1 = xq+2 = . . = x|A1 | = 0. Leaving the fixed variables as they are, create r−p new shortest path problems that must satisfy the additional conditions xp+1 = 0, xp+1 = 1, xp+2 = 0, .. xp+1 = xp+2 = . . = xr−1 = 1, xr = 0. 4 A block sequential heuristic (BLOSH) While, as we shall see in the next section, the three MIP reformulations and the exact multipath algorithm allow to tackle medium size problems, the NPhard nature of TOP will ultimately limit the size of problems that can be solved to prove optimality.
22) 0 Hence, condition (21) implies that the vector F(x ) lies within the interior of the dual to the recession cone of the set X. 1 ([24]) The variational inequality problem: to find a vector z ∈ X such that ∀x ∈ X, (23) (x − z)T F(z) ≥ 0 has a non-empty, compact, convex solution set. Proof. It is well-known [27] that the pseudo-monotonicity (20) and continuity of the mapping G imply convexity of the solution set Z = {z ∈ X : (x − z)T F(z) ≥ 0 ∀x ∈ X}, (24) of problem (23) provided that the latter is non-empty.
Since G is x) ≥ 0 for each z ∈ Z, continuous, the following limit relation holds: (z − x ¯)T G(¯ which means that x ¯ solves (29). Thus we have proved that every limit point of the generalized sequence Q solves BVI (23), (24), (29). Hence, Q can have at most one limit point. 2, it suffices to establish that the set Q is bounded, and consequently, the limit point exists. In order to do that, consider a norm-divergent sequence {xεk } of solutions to parametric problem (31) where εk → 0 as k → ∞. Without loss of generality, suppose (xεk − x0 ) → s ∈ Rn , s = 1; here x0 is the that xεk = x0 for each k, and xεk − x0 vector from condition (21).