Related papers: A seven-point algorithm for piecewise smooth univa…
We consider the problem of minimizing a convex function over the intersection of finitely many simple sets which are easy to project onto. This is an important problem arising in various domains such as machine learning. The main difficulty…
Gradient descent method, as one of the major methods in numerical optimization, is the key ingredient in many machine learning algorithms. As one of the most fundamental way to solve the optimization problems, it promises the function value…
Low rank matrix approximation is a popular topic in machine learning. In this paper, we propose a new algorithm for this topic by minimizing the least-squares estimation over the Riemannian manifold of fixed-rank matrices. The algorithm is…
We propose a new algorithm to solve optimization problems of the form $\min f(X)$ for a smooth function $f$ under the constraints that $X$ is positive semidefinite and the diagonal blocks of $X$ are small identity matrices. Such problems…
In this work, we consider convex optimization problems with smooth objective function and nonsmooth functional constraints. We propose a new stochastic gradient algorithm, called Stochastic Halfspace Approximation Method (SHAM), to solve…
We present an algorithm for approximating a function defined over a $d$-dimensional manifold utilizing only noisy function values at locations sampled from the manifold with noise. To produce the approximation we do not require any…
This paper improves the algorithms based on supporting halfspaces and quadratic programming for convex set intersection problems in our earlier paper in several directions. First, we give conditions so that much smaller quadratic programs…
This paper proposes a polynomial-time algorithm to construct the monotone stepwise curve that minimizes the sum of squared errors with respect to a given cloud of data points. The fitted curve is also constrained on the maximum number of…
It is shown how piecewise differentiable functions $F: \mathbb R^n \mapsto \mathbb R^m $ that are defined by evaluation programs can be approximated locally by a piecewise linear model based on a pair of sample points $\check x$ and $\hat…
A sequential piecewise linear programming method is presented where bounded domains of non-convex functions are successively contracted about the solution of a piecewise linear program at each iteration of the algorithm. Although…
We prove lower bounds on the error incurred when approximating any oscillating function using piecewise polynomial spaces. The estimates are explicit in the polynomial degree and have optimal dependence on the meshwidth and frequency when…
The problem of computing functions of values at the nodes in a network in a totally distributed manner, where nodes do not have unique identities and make decisions based only on local information, has applications in sensor, peer-to-peer,…
We consider the global minimization of smooth functions based solely on function evaluations. Algorithms that achieve the optimal number of function evaluations for a given precision level typically rely on explicitly constructing an…
The proximal point algorithm is a widely used tool for solving a variety of convex optimization problems such as finding zeros of maximally monotone operators, fixed points of nonexpansive mappings, as well as minimizing convex functions.…
Given noisy data, function estimation is considered when the unknown function is known apriori to consist of a small number of regions where the function is either convex or concave. When the regions are known apriori, the estimate is…
We discuss non-Euclidean deterministic and stochastic algorithms for optimization problems with strongly and uniformly convex objectives. We provide accuracy bounds for the performance of these algorithms and design methods which are…
We have devised a simple numerical technique to treat rugged data points that arise due to the insufficient gain setting error (or quantization error) of a digital instrument. This is a very wide spread problem that all experimentalists…
A heuristic formula for 5-point approximation of the first derivative of an unknown function whose values are measured with an error at unequally spaced points is proposed. The derivative at a given point is calculated using the effective…
This work is devoted to solving the composite optimization problem with the mixture oracle: for the smooth part of the problem, we have access to the gradient, and for the non-smooth part, only the one-point zero-order oracle is available.…
This paper presents smoothing schemes for obtaining approximate stationary points of unconstrained or linearly-constrained composite nonconvex-concave min-max (and hence nonsmooth) problems by applying well-known algorithms to composite…