Javier F. Peña 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
We show that a combination of two simple preprocessing steps would generally improve the conditioning of a homogeneous system of linear inequalities. Our approach is based on a comparison among three different but related notions of conditioning for linear inequalities. ArticleDownload View PDF
The perceptron algorithm, introduced in the late fifties in the machine learning community, is a simple greedy algorithm for finding a solution to a finite set of linear inequalities. The algorithm’s main advantages are its simplicity and noise tolerance. The algorithm’s main disadvantage is its slow convergence rate. We propose a modified version of the … Read more
Consider a homogeneous multifold convex conic system $$ Ax = 0, \; x\in K_1\times \cdots \times K_r $$ and its alternative system $$ A\transp y \in K_1^*\times \cdots \times K_r^*, $$ where $K_1,\dots, K_r$ are regular closed convex cones. We show that there is canonical partition of the index set $\{1,\dots,r\}$ determined by certain complementarity … Read more
Certificates of non-negativity are fundamental tools in optimization. A “certificate” is generally understood as an expression that makes the non-negativity of the function in question evident. Some classical certificates of non-negativity are Farkas Lemma and the S-lemma. The lift-and-project procedure can be seen as a certificate of non-negativity for affine functions over the union of … Read more
We propose new stochastic gradient algorithms for solving convex composite optimization problems. In each iteration, our algorithms utilize a stochastic oracle of the gradient of the smooth component in the objective function. Our algorithms are based on a stochastic version of the estimate sequence technique introduced by Nesterov (Introductory Lectures on Convex Optimization: A Basic … Read more
We provide a closed-form solution to the problem of computing the sharpest static-arbitrage upper bound on the price of a European basket option, given the prices of vanilla call options in the underlying securities. Unlike previous approaches to this problem, our solution technique is entirely based on linear programming. This also allows us to obtain … Read more
We propose an iterated version of Nesterov’s first-order smoothing method for the two-person zero-sum game equilibrium problem $$\min_{x\in Q_1} \max_{y\in Q_2} \ip{x}{Ay} = \max_{y\in Q_2} \min_{x\in Q_1} \ip{x}{Ay}.$$ This formulation applies to matrix games as well as sequential games. Our new algorithmic scheme computes an $\epsilon$-equilibrium to this min-max problem in $\Oh(\kappa(A) \ln(1/\epsilon))$ first-order iterations, … Read more
We develop first-order smoothing techniques for saddle-point problems that arise in the Nash equilibria computation of sequential games. The crux of our work is a construction of suitable prox-functions for a certain class of polytopes that encode the sequential nature of the games. An implementation based on our smoothing techniques computes approximate Nash equilibria for … Read more