The speed optimization problem over a path aims to find a set of speeds over each arc of the given path to minimize the total cost, while respecting the time-window constraint at each node and speed limits over each arc. In maritime transportation, the cost represents fuel cost or emissions, so study of this problem … Read more
We design and analyze a novel gradient-based algorithm for unconstrained convex optimization. When the objective function is $m$-strongly convex and its gradient is $L$-Lipschitz continuous, the iterates and function values converge linearly to the optimum at rates $\rho$ and $\rho^2$, respectively, where $\rho = 1-\sqrt{m/L}$. These are the fastest known guaranteed linear convergence rates for … Read more
If K is a closed convex cone and if L is a linear operator having L(K) a subset of K, then L is a positive operator on K and L preserves inequality with respect to K. The set of all positive operators on K is denoted by pi(K). If J is the dual of K, … Read more
Generalized matrix-fractional (GMF) functions are a class of matrix support functions introduced by Burke and Hoheisel as a tool for unifying a range of seemingly divergent matrix optimization problems associated with inverse problems, regularization and learning. In this paper we dramatically simplify the support function representation for GMF functions as well as the representation of … Read more
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
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
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
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