Constrained Nonlinear Optimization

Mathematical Programs with Equilibrium Constraints: A sequential optimality condition, new constraint qualifications and algorithmic consequences.

Categories Constrained Nonlinear Optimization

Mathematical programs with equilibrium (or complementarity) constraints, MPECs for short, are a difficult class of constrained optimization problems. The feasible set has a very special structure and violates most of the standard constraint qualifications (CQs). Thus, the Karush-Kuhn-Tucker (KKT) conditions are not necessarily satisfied by minimizers and the convergence assumptions of many methods for solving … Read more

A predictor-corrector path-following algorithm for dual-degenerate parametric optimization problems

Categories Constrained Nonlinear Optimization, Nonlinear Optimization Tags , , ,

Most path-following algorithms for tracing a solution path of a parametric nonlinear optimization problem are only certifiably convergent under strong regularity assumptions about the problem functions, in particular, the linear independence of the constraint gradients at the solutions, which implies a unique multiplier solution for every nonlinear program. In this paper we propose and prove … Read more

A new algebraic analysis to linear mixed models

Categories Constrained Nonlinear Optimization, Statistics Tags , , , , , ,

This article presents a new investigation to the linear mixed model $\by = \bX \bbe + \bZ\bga + \bve$ with fixed effect $\bX\bbe$ and random effect $\bZ\bga$ under a general assumption via some novel algebraic tools in matrix theory, and reveals a variety of deep and profound properties hidden behind the linear mixed model. We … Read more

A Second-Order Cone Based Approach for Solving the Trust Region Subproblem and Its Variants

Categories Constrained Nonlinear Optimization, Quadratic Programming, Second-Order Cone Programming Tags , , , ,

We study the trust region subproblem (TRS) of minimizing a nonconvex quadratic function over the unit ball with additional conic constraints. Despite having a nonconvex objective, it is known that the TRS and a number of its variants are polynomial-time solvable. In this paper, we follow a second-order cone based approach to derive an exact … Read more

Iteration-complexity of a Rockafellar’s proximal method of multipliers for convex programming based on second-order approximations

Categories Constrained Nonlinear Optimization, Convex Optimization, Global Optimization Tags , , , ,

This paper studies the iteration-complexity of a new primal-dual algorithm based on Rockafellar’s proximal method of multipliers (PMM) for solving smooth convex programming problems with inequality constraints. In each step, either a step of Rockafellar’s PMM for a second-order model of the problem is computed or a relaxed extragradient step is performed. The resulting algorithm … Read more

Numerical Solution of Linear-Quadratic Optimal Control Problems for Switching System

Categories Applications - OR and Management Sciences, Constrained Nonlinear Optimization, Control Applications Tags , , ,

In this paper we obtained an approach to the optimal switching control problem with unknown switching points which it is described in reference [1, 2]. In reference [1], the authors studied the Decomposition of Linear-Quadratic Optimal Control Problems for Two-Steps Systems. In [1], the authors assumed the switching point t1 is xed in the interval … Read more

A Riemannian rank-adaptive method for low-rank optimization

Categories Constrained Nonlinear Optimization, Data-Mining

This paper presents an algorithm that solves optimization problems on a matrix manifold $\mathcal{M} \subseteq \mathbb{R}^{m \times n}$ with an additional rank inequality constraint. The algorithm resorts to well-known Riemannian optimization schemes on fixed-rank manifolds, combined with new mechanisms to increase or decrease the rank. The convergence of the algorithm is analyzed and a weighted … Read more

A multiplier method with a class of penalty functions for convex programming

Categories Constrained Nonlinear Optimization, Nonlinear Optimization Tags , ,

We consider a class of augmented Lagrangian methods for solving convex programming problems with inequality constraints. This class involves a family of penalty functions and specific values of parameters $p,q,\tilde y \in R$ and $c>0$. The penalty family includes the classical modified barrier and the exponential function. The associated proximal method for solving the dual … Read more

Improved pointwise iteration-complexity of a regularized ADMM and of a regularized non-Euclidean HPE framework

Categories Constrained Nonlinear Optimization, Convex Optimization Tags , , , , , , ,

This paper describes a regularized variant of the alternating direction method of multipliers (ADMM) for solving linearly constrained convex programs. It is shown that the pointwise iteration-complexity of the new method is better than the corresponding one for the standard ADMM method and that, up to a logarithmic term, is identical to the ergodic iteration-complexity … Read more