We generalize the Nelder-Mead simplex and LTMADS algorithms and, the frame based methods for function minimization to Riemannian manifolds. Examples are given for functions defined on the special orthogonal Lie group $\mathcal{SO}(n)$ and the Grassmann manifold $\mathcal{G}(n,k)$. Our main examples are applying the generalized LTMADS algorithm to equality constrained optimization problems and, to the Whitney … Read more
We consider the problem of locating a single radiating source from several noisy measurements using a maximum likelihood (ML) criteria. The resulting optimization problem is nonconvex and nonsmooth and thus finding its global solution is in principal a hard task. Exploiting the special structure of the objective function, we introduce and analyze two iterative schemes … Read more
We study the problem of optimizing nonlinear objective functions over matroids presented by oracles or explicitly. Such functions can be interpreted as the balancing of multi-criteria optimization. We provide a combinatorial polynomial time algorithm for arbitrary oracle-presented matroids, that makes repeated use of matroid intersection, and an algebraic algorithm for vectorial matroids. Our work is … Read more
We present a new matrix-free method for the trust region subproblem, assuming that the approximate Hessian is updated by the limited memory BFGS formula with m = 2. The resulting updating scheme, called 2-BFGS, give us the ability to determine via simple formulas the eigenvalues of the resulting approximation. Thus, at each iteration, we can … Read more
An algorithm for solving large-scale nonlinear optimization problems with simple bounds is described. The algorithm is an $\ell_\infty$-norm trust-region method that uses both active set identification techniques as well as limited memory BFGS updating for the Hessian approximation. The trust-region subproblems are solved using primal-dual interior point techniques that exploit the structure of the limited … Read more
We present fourteen basic FORTRAN subroutines for large-scale unconstrained and box constrained optimization and large-scale systems of nonlinear equations. Subroutines {\tt PLIS} and {\tt PLIP}, intended for dense general optimization problems, are based on limited-memory variable metric methods. Subroutine {\tt PNET}, also intended for dense general optimization problems, is based on an inexact truncated Newton … Read more
Many problems in signal processing and statistical inference involve finding sparse solutions to under-determined, or ill-conditioned, linear systems of equations. A standard approach consists in minimizing an objective function which includes a quadratic (squared $\ell_2$) error term combined with a sparseness-inducing ($\ell_1$) regularization term.{\it Basis pursuit}, the {\it least absolute shrinkage and selection operator} (LASSO), … Read more
We consider a rather general class of mathematical programming problems with data uncertainty, where the uncertainty set is represented by a system of convex inequalities. We prove that the robust counterparts of this class of problems can be equivalently reformulated as finite and explicit optimization problems. Moreover, we develop simplified reformulations for problems with uncertainty … Read more
In this paper, we consider a primal-dual interior point method for solving nonlinear semidefinite programming problems. By combining the primal barrier penalty function and the primal-dual barrier function, a new primal-dual merit function is proposed within the framework of the line search strategy. We show the global convergence property of our method. Finally some numerical … Read more
In this paper, we propose a new trust-region SQP method, which uses no penalty function, for solving nonlinearly constrained optimization problem. Our method consists of alternate two phases. Specifically, we alternately proceed the feasibility restoration phase and the objective function minimization phase. The global convergence property of the proposed method is shown. Citation Cooperative Research … Read more