In this paper we consider the following two types of complex quadratic maximization problems: (i) maximize $z^{\HH} Q z$, subject to $z_k^m=1$, $k=1,…,n$, where $Q$ is a Hermitian matrix with $\tr Q=0$ and $z\in \C^n$ is the decision vector; (ii) maximize $\re y^{\HH}Az$, subject to $y_k^m=1$, $k=1,…,p$, and $z_l^m=1$, $l=1,…,q$, where $A\in \C^{p\times q}$ and … Read more
This paper studies the possibilities of the Linear Matrix Inequality (LMI) characterization of the matrix cones formed by nonnegative complex Hermitian quadratic functions over specific domains in the complex space. In its real case analog, such studies were conducted in Sturm and Zhang in 2003. In this paper it is shown that stronger results can … Read more
In the paper a primal-infeasible interior point algorithm is proposed for linearly constrained convex programming. The starting point is any positive primal-infeasible dual-feasible point in a large region. The method maintains positivity of the iterates which point satisfies primal-infeasible dual-feasible point. At each iterates it requires to solve approximately a nonlinear system. It is shown … Read more
We address a problem of covariance selection, where we seek a trade-off between a high likelihood against the number of non-zero elements in the inverse covariance matrix. We solve a maximum likelihood problem with a penalty term given by the sum of absolute values of the elements of the inverse covariance matrix, and allow for … Read more
A standard quadratic optimization problem (StQP) consists in minimizing a quadratic form over a simplex. A number of problems can be transformed into a StQP, including the general quadratic problem over a polytope and the maximum clique problem in a graph. In this paper we present several polynomial-time bounds for StQP ranging from very simple … Read more
The Hartree-Fock method is well known in quantum chemistry, and widely used to obtain atomic and molecular eletronic wave functions, based on the minimization of a functional of the energy. This gives rise to a multi-extremal, nonconvex, polynomial optimization problem. We give a novel mathematical programming formulation of the problem, which we solve by using … Read more
A theory of globally convergent trust-region methods for self-consistent field electronic structure calculations that use the density matrices as variables is developed. The optimization is performed by means of sequential global minimizations of a quadratic model of the true energy. The global minimization of this quadratic model, subject to the idempotency of the density matrix … Read more
This paper describes Knitro 5.0, a C-package for nonlinear optimization that combines complementary approaches to nonlinear optimization to achieve robust performance over a wide range of application requirements. The package is designed for solving large-scale, smooth nonlinear programming problems, and it is also effective for the following special cases: unconstrained optimization, nonlinear systems of equations, … Read more
We propose a Greedy Randomized Adaptive Search Procedure (GRASP) for solving continuous global optimization problems subject to box constraints. The method was tested on benchmark functions and the computational results show that our approach was able to find, in a few seconds, optimal solutions for all tested functions despite not using any gradient information about … Read more
Support set invariancy sensitivity analysis deals with finding the range of the parameter variation where there are optimal solutions with the same positive variables for all parameter values throughout this range. This approach to sensitivity analysis has been studied for Linear Optimization (LO) and Convex Quadratic Optimization (CQO) problems, when they are in standard form. … Read more