Parametric analysis of semidefinite optimization

In this paper, we study parametric analysis of semidefinite optimization problems w.r.t. the perturbation of the objective function. We study the behavior of the optimal partition and optimal set mapping on a so-called nonlinearity interval. Furthermore, we investigate the sensitivity of the approximation of the optimal partition in a nonlinearity interval, which has been recently studied by Mohammad-Nezhad and Terlaky. The approximation of the optimal partition was obtained from a bounded sequence of interior solutions on, or in a neighborhood of the central path. We derive an upper bound on the distance between the approximations of the optimal partitions of the original and perturbed problems. Finally, we examine the theoretical bounds by way of experimentation

A Monte Carlo Analysis of the VAR-Based Indirect Inference Estimation of DSGE Models

In this paper we study estimation of DSGE models. More specifically, in the indirect inference framework, we analyze how critical is the choice of the reduced form model for estimation purposes. As it turns out, simple VAR parameters performs better than commonly used impulse response functions. This can be attributed to the fact that IRF worsen identification issues for models that are already plagued by that phenomenon.

Automatic differentiation of hybrid models Illustrated by Diffedge Graphic Methodology. (Survey)

We investigate the automatic differentiation of hybrid models, viz. models that may contain delays, logical tests and discontinuities or loops. We consider differentiation with respect to parameters, initial conditions or the time. We emphasize the case of a small number of derivations and iterated differentiations are mostly treated with a foccus on high order iterations of the same derivation. The models we consider may involve arithmetic operations, elementary functions, logical tests but also more elaborate components such as delays, integrators, equations and differential equations solvers. This survey has no pretention to exhaustivity but tries to fil a gap in the litterature where each kind of of component may be documented, but seldom their common use. The general approach is illustrated by computer algebra experiments, stressing the interest of performing differentiation, whenever possible, on high level objects, before any translation in Fortran or C code. We include ordinary differential systems with discontinuity, with a special interest for those comming from discontinuous Lagrangians. We conclude with an overview of the graphic methodology developped in the Diffedge software for Simulink hybrid models. Not all possibilities are covered, but the methodology can be adapted. The result of automatic differentiation is a new block diagram and so it can be easily translated to produce real time embedded programs. We welcome any comments or suggestions of references that we may have missed.Comment: 47 p. Source files from computer experiments availabl

Using the RD rational Arnoldi method for exponential integrators

In this paper we investigate some practical aspects concerning the use of the Restricted-Denominator (RD) rational Arnoldi method for the computation of the core functions of exponential integrators for parabolic problems. We derive some useful a-posteriori bounds together with some hints for a suitable implementation inside the integrators. Numerical ex- periments arising from the discretization of sectorial operators are pre- sented.Comment: 27 pages, 5 figure

A stochastic approximation method for approximating the efficient frontier of chance-constrained nonlinear programs

We propose a stochastic approximation method for approximating the efficient frontier of chance-constrained nonlinear programs. Our approach is based on a bi-objective viewpoint of chance-constrained programs that seeks solutions on the efficient frontier of optimal objective value versus risk of constraint violation. To this end, we construct a reformulated problem whose objective is to minimize the probability of constraints violation subject to deterministic convex constraints (which includes a bound on the objective function value). We adapt existing smoothing-based approaches for chance-constrained problems to derive a convergent sequence of smooth approximations of our reformulated problem, and apply a projected stochastic subgradient algorithm to solve it. In contrast with exterior sampling-based approaches (such as sample average approximation) that approximate the original chance-constrained program with one having finite support, our proposal converges to stationary solutions of a smooth approximation of the original problem, thereby avoiding poor local solutions that may be an artefact of a fixed sample. Our proposal also includes a tailored implementation of the smoothing-based approach that chooses key algorithmic parameters based on problem data. Computational results on four test problems from the literature indicate that our proposed approach can efficiently determine good approximations of the efficient frontier

On the convergence of an algorithm constructing the normal form for lower dimensional elliptic tori in planetary systems

We give a constructive proof of the existence of lower dimensional elliptic tori in nearly integrable Hamiltonian systems. In particular we adapt the classical Kolmogorov's normalization algorithm to the case of planetary systems, for which elliptic tori may be used as replacements of elliptic keplerian orbits in Lagrange-Laplace theory. With this paper we support with rigorous convergence estimates the semi-analytical work in our previous article (2011), where an explicit calculation of an invariant torus for a planar model of the Sun-Jupiter-Saturn-Uranus system has been made. With respect to previous works on the same subject we exploit the characteristic of Lie series giving a precise control of all terms generated by our algorithm. This allows us to slightly relax the non-resonance conditions on the frequencies.Comment: 45 page

bridgesampling: An R Package for Estimating Normalizing Constants

Statistical procedures such as Bayes factor model selection and Bayesian model averaging require the computation of normalizing constants (e.g., marginal likelihoods). These normalizing constants are notoriously difficult to obtain, as they usually involve high-dimensional integrals that cannot be solved analytically. Here we introduce an R package that uses bridge sampling (Meng & Wong, 1996; Meng & Schilling, 2002) to estimate normalizing constants in a generic and easy-to-use fashion. For models implemented in Stan, the estimation procedure is automatic. We illustrate the functionality of the package with three examples

Decentralized allocation of human capital and nonlinear growth

The standard two-sector growth model with physical and human capital characterizes a process of material accumulation involving simple dynamics; constant long run growth is observable when assuming conventional Cobb-Douglas production functions in both sectors. This framework is developed under a central planner scenario: it is a representative agent that chooses between consumption and capital accumulation, on one hand, and between allocating human capital to each one of the two sectors, on the other. We concentrate in this second choice and we argue that the outcome of the aggregate model is incompatible with a scenario where individual agents, acting in a market economy, are free to decide, in each time moment, how to allocate their human capital in order to produce goods or to create additional skills. Combining individual incentives, the effort of a central planner (i.e., government) to approximate the decentralized outcome to the optimal result and a discrete choice rule that governs the decisions of individual agents, we propose a growth framework able to generate a significant variety of long term dynamic results, including endogenous fluctuations.Endogenous growth; Human capital; Endogenous business cycles; Discrete choice; Nonlinear dynamics; Chaos

The Small and Large Time Implied Volatilities in the Minimal Market Model

This paper derives explicit formulas for both the small and large time limits of the implied volatility in the minimal market model. It is shown that interest rates do impact on the implied volatility in the long run even though they are negligible in the short time limit.small and large time implied volatility; benchmark approach; square-root process; the minimal market model

A Regularized Semi-Smooth Newton Method With Projection Steps for Composite Convex Programs

The goal of this paper is to study approaches to bridge the gap between first-order and second-order type methods for composite convex programs. Our key observations are: i) Many well-known operator splitting methods, such as forward-backward splitting (FBS) and Douglas-Rachford splitting (DRS), actually define a fixed-point mapping; ii) The optimal solutions of the composite convex program and the solutions of a system of nonlinear equations derived from the fixed-point mapping are equivalent. Solving this kind of system of nonlinear equations enables us to develop second-order type methods. Although these nonlinear equations may be non-differentiable, they are often semi-smooth and their generalized Jacobian matrix is positive semidefinite due to monotonicity. By combining with a regularization approach and a known hyperplane projection technique, we propose an adaptive semi-smooth Newton method and establish its convergence to global optimality. Preliminary numerical results on $\ell_1$-minimization problems demonstrate that our second-order type algorithms are able to achieve superlinear or quadratic convergence.Comment: 25 pages, 4 figure