By Eduard L. Stiefel
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This page intentionally left blank 3 THE CENTRAL PATH Preamble If the system of necessary and sufficient optimality conditions for (P) and (D) is perturbed by introducing a parameter in a special way, then the solution of the perturbed system defines an analytic curve (parameterized by through the feasible region, which leads to the optimal set as This curve is called the central path and most interior point methods ‘follow’ the central path approximately to reach the optimal set. We will review various properties of the central path.
4 in Appendix B). This theorem essentially states that two convex sets in can be separated by a hyperplane if and only if their relative interiors are disjoint. 2 (Strong duality) Assume that (resp. assume that (D) (resp. (P)) is strictly feasible. It now holds that and Proof: We will first consider the case where is trivial if since then Further (resp. and (D) is strictly feasible. The proof is optimal for (P). We can therefore assume Let us define the (nonempty) convex set The relative interiors of and are disjoint, by construction.
In what follows, we will assume strict feasibility of (P) and (D), unless otherwise indicated. Also, the range (or column) space of any primal (resp. dual) feasible will be denoted by (resp. e. there exists an optimal solution pair such that For general SDP this is not the case, as the next example shows. 3 (Alizadeh et al. ) Let and The optimal solutions of (P) and (D) are given by The solution is clearly optimal, since and therefore It is also easy to see that the optimal solutions are unique, and therefore strict complementarity does not hold for this example.
An Introduction to Numerical Mathematics by Eduard L. Stiefel