Javier F. Peña This paper documents a computational implementation of a {\em projection and rescaling algorithm} for finding most interior solutions to the pair of feasibility problems find $x\in L\cap\mathbb{R}^n_{+} $ and find $x\in L^\perp\cap\mathbb{R}^n_{+},$ where $L$ denotes a linear subspace in $\mathbb{R}^n$ and $L^\perp$ denotes its orthogonal complement. The projection and rescaling algorithm is a recently developed … Read more
The condition number of a smooth convex function, namely the ratio of its smoothness to strong convexity constants, is closely tied to fundamental properties of the function. In particular, the condition number of a quadratic convex function is precisely the square of the diameter-to-width ratio of a canonical ellipsoid associated to the function. Furthermore, the … Read more
We give a novel proof of the $O(1/k)$ and $O(1/k^2)$ convergence rates of the proximal gradient and accelerated proximal gradient methods for composite convex minimization. The crux of the new proof is an upper bound constructed via the convex conjugate of the objective function. Citation Technical Report, Carnegie Mellon University, January 2018. Article Download View … Read more
This paper gives a unified and succinct approach to the $O(1/\sqrt{k}), O(1/k),$ and $O(1/k^2)$ convergence rates of the subgradient, gradient, and accelerated gradient methods for unconstrained convex minimization. In the three cases the proof of convergence follows from a generic bound defined by the convex conjugate of the objective function. Citation Working Paper, Carnegie Mellon … Read more
We give new insight into the Grassmann condition of the conic feasibility problem \[ x \in L \cap K \setminus\{0\}. \] Here $K\subseteq V$ is a regular convex cone and $L\subseteq V$ is a linear subspace of the finite dimensional Euclidean vector space $V$. The Grassmann condition of this problem is the reciprocal of the … Read more
It is known that the gradient descent algorithm converges linearly when applied to a strongly convex function with Lipschitz gradient. In this case the algorithm’s rate of convergence is determined by condition number of the function. In a similar vein, it has been shown that a variant of the Frank-Wolfe algorithm with away steps converges … Read more
We propose a simple {\em projection and algorithm} to solve the feasibility problem \[ \text{ find } x \in L \cap \Omega, \] where $L$ and $\Omega$ are respectively a linear subspace and the interior of a symmetric cone in a finite-dimensional vector space $V$. This projection and rescaling algorithm is inspired by previous work … Read more
The von Neumann algorithm is a simple coordinate-descent algorithm to determine whether the origin belongs to a polytope generated by a finite set of points. When the origin is in the interior of the polytope, the algorithm generates a sequence of points in the polytope that converges linearly to zero. The algorithm’s rate of convergence … Read more
Polynomial optimization encompasses a very rich class of problems in which both the objective and constraints can be written in terms of polynomials on the decision variables. There is a well stablished body of research on quadratic polynomial optimization problems based on reformulations of the original problem as a conic program over the cone of … Read more
The perceptron algorithm is a simple iterative procedure for finding a point in a convex cone $F$. At each iteration, the algorithm only involves a query of a separation oracle for $F$ and a simple update on a trial solution. The perceptron algorithm is guaranteed to find a feasible point in $F$ after $\Oh(1/\tau_F^2)$ iterations, … Read more