Strong duality and minimal representations for cone optimization

Abstract

The elegant results for strong duality and strict complementarity for linear programming, LP , can fail for cone programming over nonpolyhedral cones. One can have: unattained optimal values; nonzero duality gaps; and no primal-dual optimal pair that satisfies strict complementarity. This failure is tied to the nonclosure of sums of nonpolyhedral closed cones.We take a fresh look at known and new results for duality, optimality, constraint qualifications, and strict complementarity, for linear cone optimization problems in finite dimensions. These results include: weakest and universal constraint qualifications, CQs ; duality and characterizations of optimality that hold without any CQ ; geometry of nice and devious cones; the geometric relationships between zero duality gaps, strict complementarity, and the facial structure of cones; and, the connection between theory and empirical evidence for lack of a CQ and failure of strict complementarity.One theme is the notion of minimal representation of the cone and the constraints in order to regularize the problem and avoid both the theoretical and numerical difficulties that arise due to (near) loss of a CQ . We include a discussion on obtaining these representations efficiently. A parallel theme is the loss of strict complementarity and the corresponding theoretical and numerical difficulties that arise; a discussion on avoiding these difficulties is included. We include results and examples on the surprising theoretical connection between duality, strict complementarity, and nonclosure of sums of closed cones.Our emphasis is on results that deal with Semidefinite Programming, SDP .

A regularized strong duality for nonsymmetric semidefinite least squares problem

“…Compared with the facial reducing certificate considered in [3,13,24], our result strengthens theirs in the second part of Theorem 1 (i). Based on Theorem 1 (ii), by repeatedly defining F := F ∩ {U } ⊥ starting from F = M n + , we can find (M n + ) min .…”

“…Recent related research includes [12,13,15,19,20,24] and references therein. In this note, based on the facial structure of M n + , we first propose an algorithm for constructing the minimal cone (M n + ) min by employing FRA.…”

An Exact Duality Theory for Semidefinite Programming Based on Sums of Squares

“…Just like Ramana's extended Lagrange-Slater dual [Ra97], (D sos ) can be written down in polynomial time (and hence has polynomial size) in the bit size of the primal (assuming the latter has rational coefficients) and it guarantees that strong duality (i.e., weak duality, zero gap and dual attainment) always holds. Similarly, the facial reduction [BW81,TW] gives rise to a good duality theory of SDP. We refer the reader to [Pat00] for a unified treatment of these two constructions.…”

The Many Faces of Degeneracy in Conic Optimization

“…Preprocessing to rectify possible loss of "strict-feasibility" in the primal or the dual problems is appealing for general conic optimization as well. In contrast to linear programming, however, the area of preprocessing for conic optimization is in its infancy; see e.g., [29,138,30,107,109] and Section 1.1, below. In contrast to linear programming, numerical error makes preprocessing difficult in full generality.…”