A Simplified Treatment of Ramana's Exact Dual for Semidefinite Programming

半定规划中Ramana精确对偶的组合方法

A combinatorial approach to Ramana's exact dual for semidefinite programming

本文提出一种组合方法，更简单且更完整地处理Ramana半定规划精确对偶，完全刻画其可行集，并证明严格可行性蕴含原始问题仅零矩阵可行，从而给出简短透明的推导。

Strong duality in conic linear programming: facial reduction and extended duals

The facial reduction algorithm of Borwein and Wolkowicz and the extended dual of Ramana provide a strong dual for the conic linear program $$ (P) \sup {<c, x> | Ax \leq_K b} $$ in the absence of any constraint qualification. The facial reduction algorithm solves a sequence of auxiliary optimization problems to obtain such a dual. Ramana's dual is applicable when (P) is a semidefinite program (SDP) and is an explicit SDP itself. Ramana, Tuncel, and Wolkowicz showed that these approaches are closely related; in particular, they proved the correctness of Ramana's dual using certificates from a fa…

Strong duality for a problem of linear copositive programming

The paper is dedicated to the study of strong duality for a problem of linear copositive programming. Based on the recently introduced concept of the set of normalized immobile indices, an extended dual problem is deduced. The dual problem satisfies the strong duality relations and does not require any additional regularity assumptions such as constraint qualifications. The main difference with the previously obtained results consists in the fact that now the extended dual problem uses neither the immobile indices themselves nor the explicit information about the convex hull of these indices. …

Partial facial reduction: simplified, equivalent SDPs via approximations of the PSD cone

We develop a practical semidefinite programming (SDP) facial reduction procedure that utilizes computationally efficient approximations of the positive semidefinite cone. The proposed method simplifies SDPs with no strictly feasible solution (a frequent output of parsers) by solving a sequence of easier optimization problems and could be a useful pre-processing technique for SDP solvers. We demonstrate effectiveness of the method on SDPs arising in practice, and describe our publicly-available software implementation. We also show how to find maximum rank matrices in our PSD cone approximation…

Inexactness of the Hydro-Thermal Coordination Semidefinite Relaxation

Hydro-thermal coordination is the problem of determining the optimal economic dispatch of hydro and thermal power plants over time. The physics of hydroelectricity generation is commonly simplified in the literature to account for its fundamentally nonlinear nature. Advances in convex relaxation theory have allowed the advent of Shor's semidefinite programming (SDP) relaxations of quadratic models of the problem. This paper shows how a recently published SDP relaxation is only exact if a very strict condition regarding turbine efficiency is observed, failing otherwise. It further proposes the …

AC Transmission Network Expansion Planning: A Semidefinite Programming Branch-and-Cut Approach

Transmission network expansion planning is a mixed-integer optimization problem, whose solution is used to guide future investment in transmission equipment. An approach is presented to find the global solution of the transmission planning problem using an AC network model. The approach builds on the semidefinite relaxation of the AC optimal power flow problem (ACOPF); its computational engine is a new specialized branch-and-cut algorithm for transmission expansion planning to deal with the underlying mixed-integer ACOPF problem. Valid inequalities that are based on specific knowledge of the e…