The optimal reactive power dispatch (ORPD) problem is an alternating current optimal power flow (ACOPF) problem where discrete control devices for regulating the reactive power, such as shunt elements and tap changers, are considered. The ORPD problem is modelled as a mixed-integer nonlinear optimization problem and its complexity is increased compared to the ACOPF problem, … Read more
We point out that Chubanov’s oracle-based algorithm for linear programming [5] can be applied almost as it is to linear semi-infinite programming (LSIP). In this note, we describe the details and prove the polynomial complexity of the algorithm based on the real computation model proposed by Blum, Shub and Smale (the BSS model) which is … Read more
The implementation of a linear programming interior point solver is described that is based on iterative linear algebra. The linear systems are preconditioned by a basis matrix, which is updated from one interior point iteration to the next to bound the entries in a certain tableau matrix. The update scheme is based on simplex-type pivot … Read more
This paper is concerned with a cutting-plane algorithm for solving mixed-integer semidefinite optimization (MISDO) problems. In this algorithm, the positive semidefinite constraint is relaxed, and the resultant mixed-integer linear optimization problem is repeatedly solved with valid inequalities for the relaxed constraint. We prove convergence properties of the algorithm. Moreover, to speed up the computation, we … Read more
We study decision rule approximations for generic multi-stage robust linear optimization problems. We consider linear decision rules for the case when the objective coefficients, the recourse matrices, and the right-hand sides are uncertain, and consider quadratic decision rules for the case when only the right-hand sides are uncertain. The resulting optimization problems are NP-hard but … Read more
A mixed-integer convex (MI-convex) optimization problem is one that becomes convex when all integrality constraints are relaxed. We present a branch-and-bound LP outer approximation algorithm for an MI-convex problem transformed to MI-conic form. The polyhedral relaxations are refined with K* cuts} derived from conic certificates for continuous primal-dual conic subproblems. Under the assumption that all … Read more
We study time-varying semidefinite programs (TV-SDPs), which are semidefinite programs whose data (and solutions) are functions of time. Our focus is on the setting where the data varies polynomially with time. We show that under a strict feasibility assumption, restricting the solutions to also be polynomial functions of time does not change the optimal value … Read more
We propose an efficient algorithm to solve positive a semidefinite matrix approximation problem with a trace constraint. Without constraints, it is well known that positive semidefinite matrix approximation problem can be easily solved by one-time eigendecomposition of a symmetric matrix. In this paper, we confirmed that one-time eigendecomposition is also sufficient even if a trace … Read more
Given a point x inside a polytope P and a direction d, the projection of x along d asks to find the maximum step length t such that x+td is feasible; we say x+td is a pierce point because it belongs to the boundary of P. We address this projection sub-problem with arbitrary interior points … Read more
Motivated by Chubanov’s projection-based method for linear feasibility problems [Chubanov2015], a projection and rescaling algorithm for the conic feasibility problem \[ find \; x\in L\bigcap \Omega \] is proposed in [Pena2016], where $L$ and $\Omega$ are respectively a linear subspace and the interior of a symmetric cone in a finitely dimensional vector space $V$. When … Read more