In this paper, we study parametric analysis of semidefinite optimization problems with respect to the perturbation of objective function. We investigate the behavior of the optimal partition and optimal set mapping in a so called nonlinearity interval. Furthermore, we investigate the sensitivity of the approximation of the optimal partition, which have been recently studied by Mohammad-Nezhad and Terlaky. The approximation of the optimal partition was obtained from a bounded sequence of interior solutions on, or in a neighborhood of the central path. We derive an upper bound on the distance between the invariant subspaces spanned by the approximation of the optimal partition.
Adler and Monteiro @cite_34 studied the parametric analysis of LO problems using the concept of optimal partition. Another treatment of sensitivity analysis for LO based on the optimal partition approach was given by @cite_15 and Greenberg @cite_3 . @cite_12 @cite_33 extended the optimal partition approach to linearly constrained quadratic optimization (LCQO) with perturbation in the right hand side vector and showed that the optimal value function is convex and piecewise quadratic. There have been further studies on optimal partition and parametric analysis of conic optimization problems. In contrast to LO, the optimal partition of SDO is defined as a 3-tuple of mutually orthogonal subspaces of @math , see . Goldfarb and Scheinberg @cite_22 considered a parametric SDO problem, where the objective is perturbed along a fixed direction. They derived auxiliary problems to compute the directional derivatives of the optimal value function and the so-called invariancy set of the optimal partition. Yildirim @cite_18 extended the concept of the optimal partition and the auxiliary problems in @cite_22 for linear conic optimization problems.
@cite_18 @cite_33 @cite_22 @cite_3 @cite_15 @cite_34 @cite_12
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We study convex conic optimization problems in which the right-hand side and the cost vectors vary linearly as functions of a scalar parameter. We present a unifying geometric framework that subsumes the concept of the optimal partition in linear programming (LP) and semidefinite programming (SDP) and extends it to conic optimization. Similar to the optimal partition approach to sensitivity analysis in LP and SDP, the range of perturbations for which the optimal partition remains constant can be computed by solving two conic optimization problems. Under a weaker notion of nondegeneracy, this range is simply given by a minimum ratio test. We discuss briefly the properties of the optimal value function under such perturbations. In this chapter we describe the optimal set approach for sensitivity analysis for LP. We show that optimal partitions and optimal sets remain constant between two consecutive transition-points of the optimal value function. The advantage of using this approach instead of the classical approach (using optimal bases) is shown. Moreover, we present an algorithm to compute the partitions, optimal sets and the optimal value function. This is a new algorithm and uses primal and dual optimal solutions. We also extend some of the results to parametric quadratic programming, and discuss differences and resemblances with the linear programming case. We study the approximability of multiway partitioning problems, examples of which include Multiway Cut, Node-weighted Multiway Cut, and Hypergraph Multiway Cut. We investigate these problems from the point of view of two possible generalizations: as Min-CSPs, and as Submodular Multiway Partition problems. These two generalizations lead to two natural relaxations that we call respectively the Local Distribution LP, and the Lovasz relaxation. The Local Distribution LP is generally stronger than the Lovasz relaxation, but applicable only to Min-CSP with predicates of constant size. The relaxations coincide in some cases such as Multiway Cut where they are both equivalent to the CKR relaxation. We show that the Lovasz relaxation gives a (2--2 k)-approximation for Submodular Multiway Partition with k terminals, improving a recent 2-approximation [2]. We prove that this factor is optimal in two senses: (1) A (2--2 k -- e)-approximation for Submodular Multiway Partition with k terminals would require exponentially many value queries (in the oracle model), or imply NP = RP (for certain explicit submodular functions). (2) For Hypergraph Multiway Cut and Node-weighted Multiway Cut with k terminals, both special cases of Submodular Multiway Partition, we prove that a (2--2 k -- e)-approximation is NP-hard, assuming the Unique Games Conjecture. Both our hardness results are more general: (1) We show that the notion of symmetry gap, previously used for submodular maximization problems [19, 6], also implies hardness results for submodular minimization problems. (2) Assuming the Unique Games Conjecture, we show that the Local Distribution LP gives an optimal approximation for every Min-CSP that includes the Not-Equal predicate. Finally, we connect the two hardness techniques by proving that the integrality gap of the Local Distribution LP coincides with the symmetry gap of the multilinear relaxation (for a related instance). This shows that the appearance of the same hardness threshold for a Min-CSP and the related submodular minimization problem is not a coincidence. In this paper we consider a semidefinite programming (SDP) problem in which the objective function depends linearly on a scalar parameter. We study the properties of the optimal objective function value as a function of that parameter and extend the concept of the optimal partition and its range in linear programming to SDP. We also consider an approach to sensitivity analysis in SDP and the extension of our results to an SDP problem with a parametric right-hand side.