The algorithm proposed in this paper runs in a polynomial oracle time, i.e., in a number of arithmetic operations and calls to the separation oracle bounded by a polynomial in the number of variables and in the maximum binary size of an entry of the coefficient matrix. This algorithm is much simpler than traditional polynomial … Read more
Throughout its evolution, data envelopment analysis (DEA) has mostly relied on linear programming, particularly because of simple primal-dual relations and the existence of standard software for solving linear programs. Although also non-linear models, such as Russell measure or hyperbolic measure models, have been introduced, their use in applications has been limited mainly because of their … Read more
The goal of scenario reduction is to approximate a given discrete distribution with another discrete distribution that has fewer atoms. We distinguish continuous scenario reduction, where the new atoms may be chosen freely, and discrete scenario reduction, where the new atoms must be chosen from among the existing ones. Using the Wasserstein distance as measure … Read more
The problem of minimizing the rank of a symmetric positive semidefinite matrix subject to constraints can be cast equivalently as a semidefinite program with complementarity constraints (SDCMPCC). The formulation requires two positive semidefinite matrices to be complementary. We investigate calmness of locally optimal solutions to the SDCMPCC formulation and hence show that any locally optimal … Read more
In this work we present an extension of Chubanov’s algorithm to the case of homogeneous feasibility problems over a symmetric cone K. As in Chubanov’s method for linear feasibility problems, the algorithm consists of a basic procedure and a step where the solutions are confined to the intersection of a half-space and K. Following an … Read more
In this work we are interested in nonlinear symmetric cone problems (NSCPs), which contain as special cases nonlinear semidefinite programming, nonlinear second order cone programming and the classical nonlinear programming problems. We explore the possibility of reformulating NSCPs as common nonlinear programs (NLPs), with the aid of squared slack variables. Through this connection, we show … Read more
Using the T-algebra machinery we show that the only strictly convex homogeneous cones in R^n with n >= 3 are the 2-cones, also known as Lorentz cones or second order cones. In particular, this shows that the p-cones are not homogeneous when p is not 2, 1 < p <\infty and n >= 3, thus … Read more
The basis matrices corresponding to consecutive iterations of the simplex method only differ in a single column. This fact is commonly exploited in current LP solvers to avoid having to compute a new factorization of the basis at every iteration. Instead, a previous factorization is updated to reflect the modified column. Several methods are known … Read more
Moment-sum-of-squares hierarchies of semidefinite programs can be used to approximate the volume of a given compact basic semialgebraic set $K$. The idea consists of approximating from above the indicator function of $K$ with a sequence of polynomials of increasing degree $d$, so that the integrals of these polynomials generate a convergence sequence of upper bounds … Read more
Radiation therapy is considered to be one of important treatment protocols for cancers. Radiation therapy employs several beams of ionizing radiation to kill cancer tumors, but such irradiation also causes damage to normal tissues. Therefore, a treatment plan should satisfy dose-volume constraints (DVCs). Intensity-modulated radiotherapy treatment (IMRT) enables to control the beam intensities and gives … Read more