We consider a property of positive polynomials on a compact set with a small perturbation. When applied to a Polynomial Optimization Problem (POP), the property implies that the optimal value of the corresponding SemiDefinite Programming (SDP) relaxation with sufficiently large relaxation order is bounded from below by $(f^¥ast – ¥epsilon)$ and from above by $f^¥ast … Read more
This paper presents optimal scaling of the alternating directions method of multipliers (ADMM) algorithm for a class of distributed quadratic programming problems. The scaling corresponds to the ADMM step-size and relaxation parameter, as well as the edge-weights of the underlying communication graph. We optimize these parameters to yield the smallest convergence factor of the algorithm. … Read more
The worst-case evaluation complexity of finding an approximate first-order critical point using gradient-related non-monotone methods for smooth nonconvex and unconstrained problems is investigated. The analysis covers a practical linesearch implementation of these popular methods, allowing for an unknown number of evaluations of the objective function (and its gradient) per iteration. It is shown that this … Read more
The Rayleigh-Ritz (RR) procedure, including orthogonalization, constitutes a major bottleneck in computing relatively high dimensional eigenspaces of large sparse matrices. Although operations involved in RR steps can be parallelized to a certain level, their parallel scalability, which is limited by some inherent sequential steps, is lower than dense matrix-matrix multiplications. The primary motivation of this … Read more
Faster, cheaper, and more power efficient optimization solvers than those currently offered by general-purpose solutions are required for extending the use of model predictive control (MPC) to resource-constrained embedded platforms. We propose several custom computational architectures for different first-order optimization methods that can handle linear-quadratic MPC problems with input, input-rate, and soft state constraints. We … Read more
This paper studies an extended trust region subproblem (eTRS)in which the trust region intersects the unit ball with m linear inequality constraints. When m=0, m=1, or m=2 and the linear constraints are parallel, it is known that the eTRS optimal value equals the optimal value of a particular convex relaxation, which is solvable in polynomial … Read more
We address the following generalization $P$ of the Lowner-John’s ellipsoid problem. Given a (non necessarily convex) compact set $K\subset R^n$ and an even integer $d, find an homogeneous polynomial $g$ of degree $d$ such that $K\subset G:=\{x:g(x)\leq1\}$ and $G$ has minimum volume among all such sets. We show that $P$ is a convex optimization problem … Read more
In this paper we study a class of derivative-free unconstrained minimization algorithms employing nonmonotone inexact linesearch techniques along a set of suitable search directions. In particular, we define globally convergent nonmonotone versions of some well-known derivative-free methods and we propose a new algorithm combining coordinate rotations with approximate simplex gradients. Through extensive numerical experimentation, we … Read more
In this paper, we consider an interior point method for nonlinear semidefinite programming. Yamashita, Yabe and Harada presented a primal-dual interior point method in which a nondifferentiable merit function was used. By using shifted barrier KKT conditions, we propose a differentiable primal-dual merit function within the framework of the line search strategy, and prove the … Read more
We present a new algorithm for the solution of Generalized Nash Equilibrium Problems. This hybrid method combines the robustness of a potential reduction algorithm and the local quadratic convergence rate of the LP-Newton method. We base our local convergence theory on an error bound and provide a new sufficient condition for it to hold that … Read more