Sign Constraints Research Articles

Bidiagonal decompositions, minors and applications

Matrices, called ε-BD matrices, that have a bidiagonal decomposition satisfying some sign constraints are analyzed. The ε-BD matrices include all nonsingular totally positive matrices, as well as their matrices opposite in sign and their inverses. The signs of minors of ε-BD matrices are analyzed. The zero patterns of ε-BD matrices and their triangular factors are studied and applied to prove the backward stability of Gaussian elimination without pivoting for the associated linear systems.

Standard bi-quadratic optimization problems and unconstrained polynomial reformulations

A so-called Standard Bi-Quadratic Optimization Problem (StBQP) consists in minimizing a bi-quadratic form over the Cartesian product of two simplices (so this is different from a Bi-Standard QP where a quadratic function is minimized over the same set). An application example arises in portfolio selection. In this paper we present a bi-quartic formulation of StBQP, in order to get rid of the sign constraints. We study the first- and second-order optimality conditions of the original StBQP and the reformulated bi-quartic problem over the product of two Euclidean spheres. Furthermore, we discuss the one-to-one correspondence between the global/local solutions of StBQP and the global/local solutions of the reformulation. We introduce a continuously differentiable penalty function. Based upon this, the original problem is converted into the problem of locating an unconstrained global minimizer of a (specially structured) polynomial of degree eight.

Power-shaping control of reaction systems: The CSTR case

Power-shaping control is a recent approach for the control of nonlinear systems based on the physics of the dynamical system. It rests on the formulation of the dynamics in the Brayton–Moser form. One of the main obstacles for using the power-shaping approach is to write the dynamics in the required form, since a partial differential equation system submitted to sign constraints has to be solved. This work comes within the framework of control design approaches that could possibly generate a closer link between the notions of energy that are specific to reaction systems as derived from thermodynamics concepts, and the dynamic system stability theory. The objective of this paper is to address the design of power-shaping control to reaction systems, and more particularly the step of solving the partial differential equation system. In order to illustrate the approach, we have selected the classical yet complex continuous stirred tank reactor (CSTR) as a case study. We show how using the power-shaping approach leads to a global Lyapunov function for the unforced exothermic CSTR. This Lyapunov function is then reshaped by means of a controller in order to stabilize the process at a desired temperature.

Data Envelopment Analysis as Nonparametric Least-Squares Regression

Data envelopment analysis (DEA) is known as a nonparametric mathematical programming approach to productive efficiency analysis. In this paper, we show that DEA can be alternatively interpreted as nonparametric least-squares regression subject to shape constraints on the frontier and sign constraints on residuals. This reinterpretation reveals the classic parametric programming model by Aigner and Chu [Aigner, D., S. Chu. 1968. On estimating the industry production function. Amer. Econom. Rev. 58 826–839] as a constrained special case of DEA. Applying these insights, we develop a nonparametric variant of the corrected ordinary least-squares (COLS) method. We show that this new method, referred to as corrected concave nonparametric least squares (C2NLS), is consistent and asymptotically unbiased. The linkages established in this paper contribute to further integration of the econometric and axiomatic approaches to efficiency analysis.

Issues in second language literacy education with learners who are deaf

Deaf learners whose first language is American Sign Language face particular challenges and constraints in developing literacy in English as a second language. These constraints are interrogated and discussed in terms of their relationship to issues of language proficiency in both L1 and L2, and to models of second language literacy education. Suggestions are proposed as to ways in which these constraints might be addressed in designing future bilingual programs for deaf learners.

Data Envelopment Analysis as Nonparametric Least Squares Regression

Data Envelopment Analysis (DEA), a nonparametric mathematical programming approach to productive efficiency analysis, envelops all observed data. In this paper we show that DEA can be interpreted as nonparametric least squares regression subject to shape constraints on frontier and sign constraints on residuals, and that classic parametric programming model is a constrained special case of DEA. Applying these insights, we present a nonparametric variant of the corrected ordinary least squares (COLS) method. We show that this new method, which we term corrected concave nonparametric least squares (C2NLS) is consistent and asymptotically unbiased. The linkages established in this paper contribute to further integration of the econometric and linear programming approaches.

Stability of the Feasible Set Mapping of Linear Systems with an Exact Constraint Set

This paper deals with the stability of the feasible set mapping of linear systems of an arbitrary number (possibly infinite) of equations and inequalities such that the variable x ranges on a certain fixed constraint set X⊂ℝ n (X could represent the solution set of a given constraint system, e.g., the positive cone of ℝ n in the case of sign constraints). More in detail, the paper provides necessary as well as sufficient conditions for the lower and upper semicontinuity (in Berge sense), and the closedness, of the set-valued mapping which associates, with any admissible perturbation of the given (nominal) system its feasible set. The parameter space is formed by all the systems having the same structure (i.e., the same number of variables, equations and inequalities) as the nominal one, and the perturbations are measured by means of the pseudometric of the uniform convergence.

On simultaneous optimization of truss geometry and topology

The paper addresses the classical problem of optimal truss design where cross-sectional areas and the positions of joints are simultaneously optimized. Se-veral approaches are discussed from a general point of view. In particular, we focus on the difference between simultaneous and alternating optimization of geometry and topology. We recall a rigorously mathematical approach based on the implicit programming technique which considers the classical single load minimum compliance problem subject to a volume constraint. This approach is refined leading to three new problem formulations which can be treated by methods of Mathematical Programming. In particular, these formulations cover the effect of melting end nodes, i.e., vanishing potential bars due to changes in the geometry. In one of these new problem formulations, the objective function is a polynomial of degree three and the constraints are bilinear or just sign constraints. Because heuristics is avoided, certain optimality properties can be proven for resulting structures. The paper closes with two numerical test examples.

Reconstruction of complex-valued electron density with x-ray in-line holograms

A reconstruction algorithm is proposed to analyze the complex-valued electron-density distribution in an object with x-ray in-line holograms. The real and imaginary parts of the electron density correspond to local variations of x-ray phase shift and absorption, respectively. In the algorithm, the least-squares error of the holograms is iteratively minimized under sign constraints on the real and imaginary parts. The constraints, which are derived from the physical conditions in the interaction between x rays and materials, facilitate robust reconstruction. The reconstruction was applied to holograms of a biological specimen that caused both phase shift and absorption.

Relevance of functional flexibility for heterogeneous sales response models: A comparison of parametric and semi-nonparametric models

So far studies estimating sales response functions on the basis of store-specific data either consider heterogeneity or functional flexibility. That is why in this contribution a model is developed possessing both these features. It is a multilayer perceptron with store-specific coefficients which follows a hierarchical Bayesian framework. An appropriate Markov Chain Monte Carlo estimation technique is introduced capable to satisfy theoretical constraints (e.g. sign constraints on elasticities). The empirical study refers to a data base consisting of weekly observations of sales and prices for nine leading brands of a packaged consumer good category. The data were acquired in 81 stores over a time span of at least 61 weeks. The multilayer perceptron is compared to a strict parametric multiplicative model and turns out to be clearly superior in terms of posterior model probability. This result indicates the benefits of using a flexible model even if heterogeneity is dealt with. Estimated sales curves and elasticities demonstrate that both models differ with regard to implications on price response.

Bayesian Adaptive Trading with a Daily Cycle

Standard models of algorithmic trading neglect the presence of a daily cycle. We construct a model in which the trader uses information from observations of price evolution during the day to continuously update his estimate of other traders9 target sizes and directions. He uses this information to determine an optimal trade schedule to minimize total expected cost of trading, subject to sign constraints (never buy as part of a sell program). We argue that although these strategies are determined using very simple dynamic reasoning—at each moment they assume that current conditions will last until the end of trading—they are in fact the globally optimal strategies as would be determined by dynamic programming. <b>TOPICS:</b>Equity portfolio management, portfolio construction, exchanges/markets/clearinghouses

On the stability of linear systems with an exact constraint set

This paper deals with the stability of the intersection of a given set \( X\subset \mathbb{R}^{n}\)with the solution, \(F\subset \mathbb{R}^{n}\), of a given linear system whose coefficients can be arbitrarily perturbed. In the optimization context, the fixed constraint set X can be the solution set of the (possibly nonlinear) system formed by all the exact constraints (e.g., the sign constraints), a discrete subset of \(\mathbb{R}^{n}\) (as \( \mathbb{Z}^{n}\) or { 0,1} n, as it happens in integer or Boolean programming) as well as the intersection of both kind of sets. Conditions are given for the intersection \(F \cap X\) to remain nonempty (or empty) under sufficiently small perturbations of the data.

Sign Constraints Improve the Detection of Differences between Complex Spectral Data Sets: LC−IR As an Example

Spectroscopy is a fast and rich analytical tool. On many occasions, spectra are acquired of two or more sets of samples that differ only slightly. These data sets then need to be compared and analyzed, but sometimes it is difficult to find the differences. We present a simple and effective method that detects and extracts new spectral features in a spectrum coming from one set with respect to spectra of another set on the basis of the fact that these new spectral features are essentially positive quantities. The proposed procedure (i) characterizes the spectra of the reference set by a component model and (ii) uses asymmetric least squares (ASLS) to find differences with respect to this component model. It should be stressed that the method only focuses on new features and does not trace relative changes of spectral features that occur in both sets of spectra. A comparison is made with the conventional ordinary least squares (OLS) approach. Both methods (OLS and ASLS) are illustrated with simulations and are tested for size-exclusion chromatography with infrared detection (SEC-IR) of mixtures of polymer standards. Both methods are able to provide information about new spectral features. It is shown that the ASLS-based procedure yields the best recovery of new features in the simulations and in the SEC-IR experiments. Band positions and band shapes of new spectral features are better retrieved with the ASLS than with the OLS method, even those which could hardly be detected visually. Depending on the spectroscopic technique used, the ASLS-based method facilitates identification of the new chemical compounds.

A modified goal programming approach for the mean-absolute deviation portfolio optimization model

The purpose of this paper is to present a reformulation of the model presented by Feinstein and Thapa [C.D. Feinstein, M.N. Thapa, Notes: a reformulation of a mean-absolute deviation portfolio optimization model, Management Science 39 (12) (1993) 1552-1553]. The approach of Feinstein and Thapa has been accepted as the most efficient technique published, requiring the least number of auxiliary constraints and additional continuous variables. To solve a portfolio optimization problem with T periods, in their method would introduce T+2 auxiliary constraints, 2T auxiliary sign constraints, and 2T additional continuous variables. This note indicates that it is still possible to reduce the number of auxiliary constraints and additional continuous variables in the model of Feinstein and Thapa. The equivalent concise model is proposed in this note, which has T+2 auxiliary constraints, T auxiliary sign constraints, and T additional continuous variables.

Deconvolution of Well-Test Data as a Nonlinear Total Least-Squares Problem

Summary We present a new time-domain method for the deconvolution of well test data which is characterized by three novel features: (1) Instead of the rate-normalized pressure derivative itself, we estimate its logarithm, which makes explicit sign constraints unnecessary; (2) the formulation accounts for errors in both rate and pressure data, and thus amounts to a Total Least Squares (TLS) problem; and (3) regularization is based on a measure of the overall curvature of its graph. The resulting separable nonlinear TLS problem is solved using the Variable Projection algorithm. A comprehensive error analysis is given. The paper also includes tests with a simulated example and an application to a large field example.

Maximum likelihood estimation of pseudo-stoichiometry in macroscopic biological reaction schemes

Identification of pseudo-stoichiometric (or yield) coefficients is of primary importance for building a bioprocess model. In most of the applications, the estimation of these coefficients has to be performed without any knowledge of the kinetics and on the basis of a few experiments for which noisy discrete measurements of component concentrations are available. This paper proposes maximum likelihood estimators which are able to deal with measurement errors on all the signals, at each sampling time (including the initial one) and with intrinsic sign constraints on the parameters. This kind of realistic hypotheses exclude the use of the usual (weighted) least-squares estimators. The maximum likelihood estimators are proved to be unbiased (provided a first-order approximation) and their estimation error covariance matrix can be computed (at the same level of first-order approximation). The solutions are proposed in a very general framework, dealing with cell cultures (of bacteria, yeasts or animal cells) performed in stirred tank (continuous, semi-batch or batch) reactors, and without any a priori knowledge on the kinetics. The use of the estimators and their statistical properties are illustrated in a simulation case study (fed-batch bacterial cultures) and in a real case one (batch animal cell cultures).

85.67 Sign constraints on the graph of f(x) = g(y)

85.67 Sign constraints on the graph of f(x) = g(y)

On the condition numbers for polyhedra in Karmarkar's form

We present formulations of two condition measures (one for linear programming (LP) due to Ye, and the other for a matrix) as optimization problems over sign constraints. We construct, based on LP duality, a dual characterization of Ye's condition measure in the setting of Karmarkar's form. The elementary formulations (utilizing the dual characterization) lead to trivial proofs of some results relating these two condition measures for polyhedra in Karmarkar's form. Such interpretations, using the sign constraints, allow for the definition of families of condition measures that are “between” the two condition measures. Our viewpoint provides further understanding of the relationship of the two condition measures. As a result of this new understanding, we point to a connection with oriented matroids and prove that a conjecture of Vavasis and Ye on the relationship of these two condition measures is false.

Structural Properties of Maximally Robust Controllers

This work contains results on properties of integrating feedback controllers which give maximal stability robustness against real coefficient uncertainty in the numerator and denominator coefficients of transfer functions of classes of linear discretetime scalar plants. The integrator imposes simply computed upper bounds on the size of the smallest destabilising plant numerator and denominator uncertainties. The ability for these bounds to be achieved for a given nominal plant depends on the existence of solutions of particular sign properties to interpolation problems. The first main result of this paper is to use these sign constraints to infer properties of the roots of corresponding controller polynomials. In particular it is shown how the robustness bound on denominator uncertainty cannot be achieved by a controller with any strictly unstable poles. The second main result concerns cancellation of nominal minimum phase plant zeros and its effect on robustness.

On the Lavrentiev Phenomenon for Optimal Control Problems with Second-Order Dynamics

The present article examines control problems in one dimension for which there is an autonomous running cost and a specified terminal state. In this case, when the running cost involves only the control and the state, it is known that the infimal cost corresponding to any initial state is unaffected by the precise choice of $L^p$ space $(1 \leq p < \infty)$ which is specified for controls to be admissible. Here we show that the situation is different in the case of an autonomous running cost involving, in addition to the control, the state and its derivative. That is, despite the density of each space with higher exponent in those with lower exponent, the infimal cost will generally depend on the choice of $p$ if sign constraints are present.