We investigate in this paper a fixed parameter polynomial algorithm for the cardinality constrained quadratic optimization problem, which is NP-hard in general. More specifically, we prove that, given a problem of size $n$, the number of decision variables, and $s$, the cardinality, if, for some $0
Penalty and interior-point methods for nonlinear optimization problems have enjoyed great successes for decades. Penalty methods have proved to be effective for a variety of problem classes due to their regularization effects on the constraints. They have also been shown to allow for rapid infeasibility detection. Interior-point methods have become the workhorse in large-scale optimization … Read more
We consider optimization problems with objective and constraint functions that may be nonconvex and nonsmooth. Problems of this type arise in important applications, many having solutions at points of nondifferentiability of the problem functions. We present a line search algorithm for situations when the objective and constraint functions are locally Lipschitz and continuously differentiable on … Read more
In this work we consider the problem $\min_x \{ f(x) + P(x) \}$, where $f$ is $\mathcal{C}^2$ and $P$ is nonsmooth, but contains an underlying smooth substructure. Specifically, we assume the function $P$ is prox-regular partly smooth with respect to a active manifold $\M$. Recent work by Tseng and Yun \cite{tseng-yun-2009}, showed that such a … Read more
We propose a novel robust optimization technique, which is applicable to nonconvex and simulation-based problems. Robust optimization finds decisions with the best worst-case performance under uncertainty. If constraints are present, decisions should also be feasible under perturbations. In the real-world, many problems are nonconvex and involve computer-based simulations. In these applications, the relationship between decision … Read more
Global optimization algorithms, e.g., spatial branch-and-bound approaches like those implemented in codes such as BARON and COUENNE, have had substantial success in tackling complicated, but generally small scale, non-convex MINLPs (i.e., mixed-integer nonlinear programs having non-convex continuous relaxations). Because they are aimed at a rather general class of problems, the possibility remains that larger instances … Read more
We describe techniques combining the S-lemma and computation of projected quadratics which experimentally yield strong bounds on the value of convex quadratic programs with nonconvex constraints Citation unpublished report, Columbia University, March 2009 Article Download View Eigenvalue techniques for proving bounds for convex objective, nonconvex programs
Proximal bundle methods have been shown to be highly successful optimization methods for unconstrained convex problems with discontinuous first derivatives. This naturally leads to the question of whether proximal variants of bundle methods can be extended to a nonconvex setting. This work proposes an approach based on generating cutting-planes models, not of the objective function … Read more
In this paper a new algorithm to locally minimize nonsmooth, nonconvex functions is developed. We introduce the notion of secants and quasisecants for nonsmooth functions. The quasisecants are applied to find descent directions of locally Lipschitz functions. We design a minimization algorithm which uses quasisecants to find descent directions. We prove that this algorithm converges … Read more
We present a line-search algorithm for large-scale continuous optimization. The algorithm is matrix-free in that it does not require the factorization of derivative matrices. Instead, it uses iterative linear system solvers. Inexact step computations are supported in order to save computational expense during each iteration. The algorithm is an interior-point approach derived from an inexact … Read more