The positive semidefinite Procrustes (PSDP) problem is the following: given rectangular matrices $X$ and $B$, find the symmetric positive semidefinite matrix $A$ that minimizes the Frobenius norm of $AX-B$. No general procedure is known that gives an exact solution. In this paper, we present a semi-analytical approach to solve the PSDP problem. First, we characterize … Read more
This paper presents a minimization method for Lipschitz continuous, piecewise smooth objective functions based on algorithmic differentiation (AD). We assume that all nondifferentiabilities are caused by abs(), min(), and max(). The optimization method generates successively piecewise linearizations in abs-normal form and solves these local subproblems by exploiting the resulting kink structure. Both, the generation of … Read more
Common numerical methods for constrained convex optimization are predicated on efficiently computing nearest points to the feasible region. The presence of a design matrix in the constraints yields feasible regions with more complex geometries. When the functional components are gauges, there is an equivalent optimization problem—the gauge dual– where the matrix appears only in the … Read more
A split feasibility formulation for the inverse problem of intensity-modulated radiation therapy (IMRT) treatment planning with dose-volume constraints (DVCs) included in the planning algorithm is presented. It involves a new type of sparsity constraint that enables the inclusion of a percentage-violation constraint in the model problem and its handling by continuous (as opposed to integer) … Read more
We study non-parametric estimation of choice models, which was introduced to alleviate unreasonable assumptions in traditional parametric models, and are prevalent in several application areas. Existing literature focuses only on the static observational setting where all of the observations are given upfront, and lacks algorithms that provide explicit convergence rate guarantees or an a priori … Read more
The alternating direction method of multipliers (ADMM) is a popular method for the separable convex programming with linear constraints, and the proximal ADMM is its important variant. Previous studies show that the relaxation factor $\gamma\in (0, \frac{1+\sqrt{5}}{2})$ by Fortin and Glowinski for the ADMM is also valid for the proximal ADMM. In this paper, we … Read more
We study the problem of computing weighted analytic center for system of linear matrix inequality constraints. The problem can be solved using the Standard Newton’s method. However, this approach requires that a starting point in the interior point of the feasible region be given or a Phase I problem be solved. We address the problem … Read more
Consensus optimization has received considerable attention in recent years. A number of decentralized algorithms have been proposed for {convex} consensus optimization. However, to the behaviors or consensus \emph{nonconvex} optimization, our understanding is more limited. When we lose convexity, we cannot hope our algorithms always return global solutions though they sometimes still do sometimes. Somewhat surprisingly, … Read more
In this paper, we develop a Parameterized Proximal Point Algorithm (P-PPA) for solving a class of separable convex programming problems subject to linear and convex constraints. The proposed algorithm is provable to be globally convergent with a worst-case $O(1/t)$ convergence rate, where $t$ is the iteration number. By properly choosing the algorithm parameters, numerical experiments … Read more
We consider image deblurring problem in the presence of impulsive noise. It is known that \emph{total variation} (TV) regularization with L1-norm penalized data fitting (TVL1 for short) works reasonably well only when the level of impulsive noise is relatively low. For high level impulsive noise, TVL1 works poorly. The reason is that all data, both … Read more