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Finding global optima in high-dimensional optimization problems is extremely challenging since the number of function evaluations required to sufficiently explore the search space increases exponentially with its dimensionality.…
Constrained non-convex optimization problems frequently arise in control applications. Solving such problems is inherently challenging, as existing methods often converge to suboptimal local minima or incur prohibitive computational costs.…
We propose a novel method for planning shortest length piecewise-linear motions through complex environments punctured with static, moving, or even morphing obstacles. Using a moment optimization approach, we formulate a hierarchy of…
A novel algorithm is proposed for segmenting an image into multiple levels using its mean and variance. Starting from the extreme pixel values at both ends of the histogram plot, the algorithm is applied recursively on sub-ranges computed…
We study the problem of partitioning a polygon into the minimum number of subpolygons using cuts in predetermined directions such that each resulting subpolygon satisfies a given width constraint. A polygon satisfies the unit-width…
Some optimization problems coming from the Differential Geometry, as for example, the minimal submanifolds problem and the harmonic maps problem are solved here via interior solutions of appropriate multitime optimal control problems.…
We study the problem of minimizing a nonnegative separable concave function over a compact feasible set. We approximate this problem to within a factor of 1+epsilon by a piecewise-linear minimization problem over the same feasible set. Our…
In this paper, we consider numerical approximations for the optimal partition problem using Lagrange multipliers. By rewriting it into constrained gradient flows, three and four steps numerical schemes based on the Lagrange multiplier…
This paper deals with speeding up the convergence of a class of two-step iterative methods for solving linear systems of equations. To implement the acceleration technique, the residual norm associated with computed approximations for each…
The problem of minimizing convex functionals of probability distributions is solved under the assumption that the density of every distribution is bounded from above and below. A system of sufficient and necessary first-order optimality…
We study the problem of partitioning a given simple polygon $P$ into a minimum number of connected polygonal pieces, each of bounded size. We describe a general technique for constructing such partitions that works for several notions of…
In this work, based on the moving-least-squares immersed boundary method, we proposed a new technique to improve the calculation of the volume force representing the body boundary. For boundary with simple geometry, we theoretically analyse…
Indexing moving objects has been extensively studied in the past decades. Moving objects, such as vehicles and mobile device users, usually exhibit some patterns on their velocities, which can be utilized for velocity-based partitioning to…
We study the smallest intersecting and enclosing ball problems in Euclidean spaces for input objects that are compact and convex. They link and unify many problems in computational geometry and machine learning. We show that both problems…
We formulate a class of velocity-free finite-particle methods for mass transport problems based on a time-discrete incremental variational principle that combines entropy and the cost of particle transport, as measured by the Wasserstein…
We study the problem of finding the Lowner-John ellipsoid, i.e., an ellipsoid with minimum volume that contains a given convex set. We reformulate the problem as a generalized copositive program, and use that reformulation to derive…
Primal and dual algorithms are developed for solving the $n$-dimensional convex optimization problem of finding the Euclidean ball of minimum radius that covers $m$ given Euclidean balls, each with a given center and radius. Each algorithm…
Free-discontinuity problems describe situations where the solution of interest is defined by a function and a lower dimensional set consisting of the discontinuities of the function. Hence, the derivative of the solution is assumed to be a…
This paper is about how to partition decision variables while decomposing a large-scale optimization problem for the best performance of distributed solution methods. Solving a large-scale optimization problem sequen- tially can be…
In big data analysis, a simple task such as linear regression can become very challenging as the variable dimension $p$ grows. As a result, variable screening is inevitable in many scientific studies. In recent years, randomized algorithms…