Constructing a quasi-concave quadratic objective function from interviewing a decision maker

A model for constructing quadratic objective functions (=utility functions) from interviewing a decision maker is considered. The interview is designed to guarantee a unique non-trivial output of the model and to enable estimating both cardinal and ordinal utility, depending on interview scenarios. The model is provided with operational restrictions for the monotonicity of the objective function (=either only growth, or only decrease in every variable) and its quasi-concavity (=convexity of the associated preference). Thereby constructing a monotonic quasi-concave quadratic objective function is reduced to a problem of non-linear programming. To support interactive editing of a quadratic objective function, the stability of the model (the continuous dependence of the output ordinal preference on the input data) is proved. In illustration, we construct a quadratic objective function of ski station customers. Then it is used to adjust prices of 10 ski stations in the south of Stuttgart.     

On the reversible quadratic centers with monotonic period function

This paper is devoted to studying the period function of the quadratic reversible centers. In this context the interesting stratum is the family of the so-called Loud's dehomogenized systems, namely

We determine several regions in the parameter plane for which the corresponding center has a monotonic period function. To this end we first show that any of these systems can be brought by means of a coordinate transformation to a potential system. Then we apply a monotonicity criterium of R. Schaaf.

     

Solving sum of quadratic ratios fractional programs via monotonic function

In this paper, a branch-reduce-bound algorithm is proposed for globally solving a sum of quadratic ratios fractional programming with nonconvex quadratic constraints. Due to its intrinsic difficulty, less work has been devoted to globally solving this problem. The proposed algorithm is based on reformulating the problem as a monotonic optimization problem, and it turns out that the optimal solution which is provided by the algorithm is adequately guaranteed to be feasible and to be close to the actual optimal solution. Convergence of the algorithm is shown and the numerical experiments are given to show the feasibility of the proposed algorithm.     

Extreme quadratic forms of a special type in n variables (n < 28)

One constructs perfect, extreme positive-definite quadratic forms of a special form in n variables for n 28, whose frames give the simplest packings of the space Rⁿ.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 144, pp. 167–172, 1985.     

Optimal objective function approximation for separable convex quadratic programming

We present an optimal piecewise-linear approximation method for the objective function of separable convex quadratic programs. The method provides guidelines on how many grid points to use and how to position them for a piecewise-linear approximation if the error induced by the approximation is to be bounded a priori.Corresponding author.     

Computing the distribution function of the ratio of quadratic forms in normal variables

Formulas are given for computing the ratio of quadratic forms in normal variables. The doubly noncentral F-distribution function is computed.Translated from Statisticheskie Metody Otsenivaniya i Proverki Gipotez, pp. 123–128, 1986.     

Estimating a general function of a quadratic function

Let be an observation from a spherically symmetric distribution with unknown location parameter . For a general non-negative function c, we consider the problem of estimating c(||x − θ||²) under the usual quadratic loss. For p ≥ 5, we give sufficient conditions for improving on the unbiased estimator γ₀ of c(||x − θ||²) by competing estimators γ _s = γ₀ + s correcting γ₀ with a suitable function s. The main condition relies on a partial differential inequality of the form k Δs + s ² ≤ 0 for a certain constant k ≠ 0. Our approach unifies, in particular, the two problems of quadratic loss estimation and confidence statement estimation and allows to derive new results for these two specific cases. Note that we formally establish our domination results (that is, with no recourse to simulation).      

Integer resource allocations with the objective function separable into pairs of variables

This paper is concerned with the problem of optimal integer allocation having a doubly concave objective function. The problem is studied with application to some model of optimal reliability. To derive the optimality conditions, we introduce the notion of integer local equilibrium, considering three of its variations. We also investigate some additional conditions under which this equilibrium is the optimal solution.An appropriate solution algorithm is presented.     

Incomplete quadratic exponential sums in several variables

We consider incomplete exponential sums in several variables of the form

     

A closed-form solution to a quadratic optimization problem in complex variables

A special quadratic optimization problem in complex variables is investigated for a closed-form solution. Two different approaches are used. The first is a direct approach which leads to a family of solutions defined in terms of arbitrary complex constants. The second is an indirect approach based on parametrizing the objective function. It leads to a specific solution, which is a member of the above family and which is shown to be bounded.     

How bad are the BFGS and DFP methods when the objective function is quadratic?

We study the use of the BFGS and DFP algorithms with step-lengths of one for minimizing quadratic functions of only two variables. The updating formulae in this case imply nonlinear three term recurrence relations between the eigenvalues of consecutive second derivative approximations, which are analysed in order to explain some gross inefficiencies that can occur. Specifically, the BFGS algorithm may require more than 10 iterations to achieve the first decimal place of accuracy, while the performance of the DFP method is far worse. The results help to explain why the DFP method is often less suitable than the BFGS algorithm for general unconstrained optimization calculations, and they show that quadratic functions provide much information about efficiency when the current vector of variables is too far from the solution for an asymptotic convergence analysis.     

On generalized quadratic equations in three prime variables

Leta ₁,a ₂ anda ₃ be any nonzero integers which are relatively prime and not all negative. In this paper, as a parallel problem of [11] for each integerk≥2, we consider the setE(X) of positive integersb≤X which satisfy the condition of congruent solubility and that the equation $$a_1 p_1^2 + a_2 p_2^2 + a_3 p_3^k = b$$ is insoluble in primesp _j. We obtain CardE(X)≤X ^1-ε. Our result extends the wellknown classical results (by Legendre and Gauss and byDavenport andHeilbronn [2]) on the equation $$x_1^2 + x_2^2 + x_3^k = b$$ in integral variablesx _j with the above bound for CardE(X) better than that in [2].     

On positive definite quadratic forms in correlatedt variables

Summary In this paper we extend Ruben's [4] result for quadratic forms in normal variables. He represented the distribution function of the quadratic form in normal variables as an infinite mixture of chi-square distribution functions. In the central case, we show that the distribution function of a quadratic form int-variables can be represented as a mixture of beta distribution functions. In the noncentral case, the distribution function presented is an infinite series in beta distribution functions. An application to quadratic discrimination is given.     

Existence criteria for cycles in two-dimensional quadratic systems

The existence of a bifurcation of simultaneously arising cycles in a neighborhood of two equilibrium states of two-dimensional quadratic systems under a variation of one scalar parameter is proved.     

一类函数的对数完全单调性

通过对参数λ,μ的讨论,主要利用函数的单调性理论,已有对数完全单调函数的性质以及幂函数的积分表达式研究了函数Gλ,μ(x)及函数[Gλ,μ(x)]-1的对数完全单调性,并在此基础上得到了一定条件下函数Gλ,μ(x)及[Gλ,μ(x)]-1对数完全单调的充要条件.     

Linear programming via a quadratic penalty function

We use quadratic penalty functions along with some recent ideas from linearl ₁ estimation to arrive at a new characterization of primal optimal solutions in linear programs. The algorithmic implications of this analysis are studied, and a new, finite penalty algorithm for linear programming is designed. Preliminary computational results are presented.Research supported by grant No. 11-0505 from the Danish Natural Science Research Council SNF.