A globally accelerated numerical method for optical tomography with continuous wave source

A new numerical method for an inverse problem for an elliptic equation with unknown potential is proposed. In this problem the point source is running along a straight line and the source-dependent Dirichlet boundary condition is measured as the data for the inverse problem. A rigorous convergence analysis shows that this method converges globally, provided that the so-called tail function is approximated well. This approximation is verified in numerical experiments, so as the global convergence. Applications to medical imaging, imaging of targets on battlefields and to electrical impedance tomography are discussed.Comment: 31 pages, 7 figure

A PAM method for computing Wasserstein barycenter with unknown supports in D2-clustering

A Wasserstein barycenter is the centroid of a collection of discrete probability distributions that minimizes the average of the $\ell_2$-Wasserstein distance. This paper concerns with the computation of a Wasserstein barycenter under the case where the support points are not pre-specified, which is known to be a severe bottleneck in the D2-clustering due to the large-scale and nonconvexity. We develop a proximal alternating minimization (PAM) method for computing an approximate Wasserstein barycenter, and provide its global convergence analysis. This method can achieve a good accuracy at a reduced computational cost when the unknown support points of the barycenter have low cardinality. Numerical comparisons with the existing representative method on synthetic and real data show that our method can yield a little better objective values within much less computing time, and the computed approximate barycenter renders a better role in the D2-clustering

MAP Inference via L2-Sphere Linear Program Reformulation

Maximum a posteriori (MAP) inference is an important task for graphical models. Due to complex dependencies among variables in realistic model, finding an exact solution for MAP inference is often intractable. Thus, many approximation methods have been developed, among which the linear programming (LP) relaxation based methods show promising performance. However, one major drawback of LP relaxation is that it is possible to give fractional solutions. Instead of presenting a tighter relaxation, in this work we propose a continuous but equivalent reformulation of the original MAP inference problem, called LS-LP. We add the L2-sphere constraint onto the original LP relaxation, leading to an intersected space with the local marginal polytope that is equivalent to the space of all valid integer label configurations. Thus, LS-LP is equivalent to the original MAP inference problem. We propose a perturbed alternating direction method of multipliers (ADMM) algorithm to optimize the LS-LP problem, by adding a sufficiently small perturbation epsilon onto the objective function and constraints. We prove that the perturbed ADMM algorithm globally converges to the epsilon-Karush-Kuhn-Tucker (epsilon-KKT) point of the LS-LP problem. The convergence rate will also be analyzed. Experiments on several benchmark datasets from Probabilistic Inference Challenge (PIC 2011) and OpenGM 2 show competitive performance of our proposed method against state-of-the-art MAP inference methods.Comment: Accepted to International Journal of Computer Visio

Computing monotone policies for Markov decision processes: a nearly-isotonic penalty approach

This paper discusses algorithms for solving Markov decision processes (MDPs) that have monotone optimal policies. We propose a two-stage alternating convex optimization scheme that can accelerate the search for an optimal policy by exploiting the monotone property. The first stage is a linear program formulated in terms of the joint state-action probabilities. The second stage is a regularized problem formulated in terms of the conditional probabilities of actions given states. The regularization uses techniques from nearly-isotonic regression. While a variety of iterative method can be used in the first formulation of the problem, we show in numerical simulations that, in particular, the alternating method of multipliers (ADMM) can be significantly accelerated using the regularization step.Comment: This work has been accepted for presentation at the 20th World Congress of the International Federation of Automatic Control, 9-14 July 201

A Discontinuous Ritz Method for a Class of Calculus of Variations Problems

This paper develops an analogue (or counterpart) to discontinuous Galerkin (DG) methods for approximating a general class of calculus of variations problems. The proposed method, called the discontinuous Ritz (DR) method, constructs a numerical solution by minimizing a discrete energy over DG function spaces. The discrete energy includes standard penalization terms as well as the DG finite element (DG-FE) numerical derivatives developed recently by Feng, Lewis, and Neilan in [Feng2013]. It is proved that the proposed DR method converges and that the DG-FE numerical derivatives exhibit a compactness property which is desirable and crucial for applying the proposed DR method to problems with more complex energy functionals. Numerical tests are provided on the classical $p$-Laplace problem to gauge the performance of the proposed DR method.Comment: 17 pages, 1 figure and 4 table

On the Global Minimization of the Value-at-Risk

In this paper, we consider the nonconvex minimization problem of the value-at-risk (VaR) that arises from financial risk analysis. By considering this problem as a special linear program with linear complementarity constraints (a bilevel linear program to be more precise), we develop upper and lower bounds for the minimum VaR and show how the combined bounding procedures can be used to compute the latter value to global optimality. A numerical example is provided to illustrate the methodology.Comment: 21 pages, 1 figur

Convergence of the tamed Euler scheme for stochastic differential equations with Piecewise Continuous Arguments under non-Lipschitz continuous coefficients

Recently, Martin Hutzenthaler pointed out that the explicit Euler method fails to converge strongly to the exact solution of a stochastic differential equation (SDE) with superlinearly growing and globally one sided Lipschitz drift coefficient. Afterwards, he proposed an explicit and easily implementable Euler method, i.e tamed Euler method, for such an SDE and showed that this method converges strongly with order of one half. In this paper, we use the tamed Euler method to solve the stochastic differential equations with piecewise continuous arguments (SEPCAs) with superlinearly growing coefficients and prove that this method is convergent with strong order one half.Comment: 16 page

Distributed Anytime MAP Inference

We present a distributed anytime algorithm for performing MAP inference in graphical models. The problem is formulated as a linear programming relaxation over the edges of a graph. The resulting program has a constraint structure that allows application of the Dantzig-Wolfe decomposition principle. Subprograms are defined over individual edges and can be computed in a distributed manner. This accommodates solutions to graphs whose state space does not fit in memory. The decomposition master program is guaranteed to compute the optimal solution in a finite number of iterations, while the solution converges monotonically with each iteration. Formulating the MAP inference problem as a linear program allows additional (global) constraints to be defined; something not possible with message passing algorithms. Experimental results show that our algorithm's solution quality outperforms most current algorithms and it scales well to large problems

A Merit Function Approach for Evolution Strategies

In this paper, we extend a class of globally convergent evolution strategies to handle general constrained optimization problems. The proposed framework handles relaxable constraints using a merit function approach combined with a specific restoration procedure. The unrelaxable constraints in our framework, when present, are treated either by using the extreme barrier function or through a projection approach. The introduced extension guaranties to the regarded class of evolution strategies global convergence properties for first order stationary constraints. Preliminary numerical experiments are carried out on a set of known test problems as well as on a multidisciplinary design optimization proble

Counterexample to global convergence of DSOS and SDSOS hierarchies

We exhibit a convex polynomial optimization problem for which the diagonally-dominant sum-of-squares (DSOS) and the scaled diagonally-dominant sum-of-squares (SDSOS) hierarchies, based on linear programming and second-order conic programming respectively, do not converge to the global infimum. The same goes for the r-DSOS and r-SDSOS hierarchies. This refutes the claim in the literature according to which the DSOS and SDSOS hierarchies can solve any polynomial optimization problem to arbitrary accuracy. In contrast, the Lasserre hierarchy based on semidefinite programming yields the global infimum and the global minimizer with the first order relaxation. We further observe that the dual to the SDSOS hierarchy is the moment hierarchy where every positive semidefinite constraint is relaxed to all necessary second-order conic constraints. As a result, the number of second-order conic constraints grows quadratically in function of the size of the positive semidefinite constraints in the Lasserre hierarchy. Together with the counterexample, this suggests that DSOS and SDSOS are not necessarily more tractable alternatives to sum-of-squares.Comment: 10 page