On Convex Quadratic Approximation

In this paper we prove the counterintuitive result that the quadratic least squares approximation of a multivariate convex function in a finite set of points is not necessarily convex, even though it is convex for a univariate convex function. This result has many consequences both for the field of statistics and optimization. We show that convexity can be enforced in the multivariate case by using semidefinite programming techniques.Convex function;least squares;quadratic interpolation;semidefinite program- ming

Machine Learning and Portfolio Optimization

The portfolio optimization model has limited impact in practice due to estimation issues when applied with real data. To address this, we adapt two machine learning methods, regularization and cross-validation, for portfolio optimization. First, we introduce performance-based regularization (PBR), where the idea is to constrain the sample variances of the estimated portfolio risk and return, which steers the solution towards one associated with less estimation error in the performance. We consider PBR for both mean-variance and mean-CVaR problems. For the mean-variance problem, PBR introduces a quartic polynomial constraint, for which we make two convex approximations: one based on rank-1 approximation and another based on a convex quadratic approximation. The rank-1 approximation PBR adds a bias to the optimal allocation, and the convex quadratic approximation PBR shrinks the sample covariance matrix. For the mean-CVaR problem, the PBR model is a combinatorial optimization problem, but we prove its convex relaxation, a QCQP, is essentially tight. We show that the PBR models can be cast as robust optimization problems with novel uncertainty sets and establish asymptotic optimality of both Sample Average Approximation (SAA) and PBR solutions and the corresponding efficient frontiers. To calibrate the right hand sides of the PBR constraints, we develop new, performance-based k-fold cross-validation algorithms. Using these algorithms, we carry out an extensive empirical investigation of PBR against SAA, as well as L1 and L2 regularizations and the equally-weighted portfolio. We find that PBR dominates all other benchmarks for two out of three of Fama-French data sets

An algorithm for constrained one-step inversion of spectral CT data

We develop a primal-dual algorithm that allows for one-step inversion of spectral CT transmission photon counts data to a basis map decomposition. The algorithm allows for image constraints to be enforced on the basis maps during the inversion. The derivation of the algorithm makes use of a local upper bounding quadratic approximation to generate descent steps for non-convex spectral CT data discrepancy terms, combined with a new convex-concave optimization algorithm. Convergence of the algorithm is demonstrated on simulated spectral CT data. Simulations with noise and anthropomorphic phantoms show examples of how to employ the constrained one-step algorithm for spectral CT data.Comment: Submitted to Physics in Medicine and Biolog

A randomized primal distributed algorithm for partitioned and big-data non-convex optimization

In this paper we consider a distributed optimization scenario in which the aggregate objective function to minimize is partitioned, big-data and possibly non-convex. Specifically, we focus on a set-up in which the dimension of the decision variable depends on the network size as well as the number of local functions, but each local function handled by a node depends only on a (small) portion of the entire optimization variable. This problem set-up has been shown to appear in many interesting network application scenarios. As main paper contribution, we develop a simple, primal distributed algorithm to solve the optimization problem, based on a randomized descent approach, which works under asynchronous gossip communication. We prove that the proposed asynchronous algorithm is a proper, ad-hoc version of a coordinate descent method and thus converges to a stationary point. To show the effectiveness of the proposed algorithm, we also present numerical simulations on a non-convex quadratic program, which confirm the theoretical results

Multivariate Convex Approximation and Least-Norm Convex Data-Smoothing

The main contents of this paper is two-fold.First, we present a method to approximate multivariate convex functions by piecewise linear upper and lower bounds.We consider a method that is based on function evaluations only.However, to use this method, the data have to be convex.Unfortunately, even if the underlying function is convex, this is not always the case due to (numerical) errors.Therefore, secondly, we present a multivariate data-smoothing method that smooths nonconvex data.We consider both the case that we have only function evaluations and the case that we also have derivative information.Furthermore, we show that our methods are polynomial time methods.We illustrate this methodology by applying it to some examples.approximation theory;convexity;data-smoothing