Related papers: Algorithms and dimensionality reductions for conti…
We introduce an efficient and scalable method for density-based multi-material topology optimization, integrating classical mirror descent techniques with point-wise polytopal design constraints. Such constraints arise naturally in this…
Seeking tighter relaxations of combinatorial optimization problems, semidefinite programming is a generalization of linear programming that offers better bounds and is still polynomially solvable. Yet, in practice, a semidefinite program is…
This paper proposes a real moment-HSOS hierarchy for complex polynomial optimization problems with real coefficients. We show that this hierarchy provides the same sequence of lower bounds as the complex analogue, yet is much cheaper to…
This paper studies a fundamental problem in convex optimization, which is to solve semidefinite programming (SDP) with high accuracy. This paper follows from the existing robust SDP-based interior point method analysis due to [Huang, Jiang,…
An analysis of high-dimensional data can offer a detailed description of a system but is often challenged by the curse of dimensionality. General dimensionality reduction techniques can alleviate such difficulty by extracting a few…
Many combinatorial optimization problems can be formulated as the search for a subgraph that satisfies certain properties and minimizes the total weight. We assume here that the vertices correspond to points in a metric space and can take…
We investigate and derive second solutions to linear homogeneous second-order difference equations using a variety of methods, in each case going beyond the purely formal solution and giving explicit expressions for the second solution. We…
In this work, we consider the low rank decomposition (SDPR) of general convex semidefinite programming problems (SDP) that contain both a positive semidefinite matrix and a nonnegative vector as variables. We develop a rank-support-adaptive…
Both unconstrained and constrained minimax single facility location problems are considered in multidimensional space with Chebyshev distance. A new solution approach is proposed within the framework of idempotent algebra to reduce the…
In this thesis, we settle the computational complexity of some fundamental questions in polynomial optimization. These include the questions of (i) finding a local minimum, (ii) testing local minimality of a point, and (iii) deciding…
In the first part of planned series of papers the formal general solutions to selection of 80 examples of different types of second order nonlinear PDEs in two independent variables with constant parameters are given. The main goal here is…
The vast majority of Dimensionality Reduction (DR) techniques rely on second-order statistics to define their optimization objective. Even though this provides adequate results in most cases, it comes with several shortcomings. The methods…
This paper aims at developing two versions of the generalized Newton method to compute not merely arbitrary local minimizers of nonsmooth optimization problems but just those, which possess an important stability property known as tilt…
The development and use of dimension reduction methods is prevalent in modern statistical literature. This paper reviews a class of dimension reduction techniques which aim to simultaneously select relevant predictors and find clusters…
Large-scale optimization problems that seek sparse solutions have become ubiquitous. They are routinely solved with various specialized first-order methods. Although such methods are often fast, they usually struggle with not-so-well…
This paper is concerned with the numerical minimization of energy functionals in Hilbert spaces involving convex constraints coinciding with a semi-norm for a subspace. The optimization is realized by alternating minimizations of the…
We shall investigate randomized algorithms for solving large-scale linear inverse problems with general regularizations. We first present some techniques to transform inverse problems of general form into the ones of standard form, then…
This paper develops new semidefinite programming (SDP) relaxation techniques for two classes of mixed binary quadratically constrained quadratic programs (MBQCQP) and analyzes their approximation performance. The first class of problem…
A power system unit commitment (UC) problem considering uncertainties of renewable energy sources is investigated in this paper, through a distributionally robust optimization approach. We assume that the first and second order moments of…
We introduce optimal algorithms for the problems of data placement (DP) and page placement (PP) in networks with a constant number of clients each of which has limited storage availability and issues requests for data objects. The objective…