Multiobjective optimization using differential evolution for real-world portfolio optimization

Portfolio optimization is an important aspect of decision-support in investment management. Realistic portfolio optimization, in contrast to simplistic mean-variance optimization, is a challenging problem, because it requires to determine a set of optimal solutions with respect to multiple objectives, where the objective functions are often multimodal and non-smooth. Moreover, the objectives are subject to various constraints of which many are typically non-linear and discontinuous. Conventional optimization methods, such as quadratic programming, cannot cope with these realistic problem properties. A valuable alternative are stochastic search heuristics, such as simulated annealing or evolutionary algorithms. We propose a new multiobjective evolutionary algorithm for portfolio optimization, which we call DEMPO??Differential Evolution for Multiobjective Portfolio Optimization. In our experimentation, we compare DEMPO with quadratic programming and another well-known evolutionary algorithm for multiobjective optimization called NSGA-II. The main advantage of DEMPO is its ability to tackle a portfolio optimization task without simplifications, while obtaining very satisfying results in reasonable runtime.     

New and old bounds for standard quadratic optimization: dominance,equivalence and incomparability

A standard quadratic optimization problem (StQP) consists in minimizing a quadratic form over a simplex. Among the problems which can be transformed into a StQP are the general quadratic problem over a polytope, and the maximum clique problem in a graph. In this paper we present several new polynomial-time bounds for StQP ranging from very simple and cheap ones to more complex and tight constructions. The main tools employed in the conception and analysis of most bounds are Semidefinite Programming and decomposition of the objective function into a sum of two quadratic functions, each of which is easy to minimize. We provide a complete diagram of the dominance, incomparability, or equivalence relations among the bounds proposed in this and in previous works. In particular, we show that one of our new bounds dominates all the others. Furthermore, a specialization of such bound dominates Schrijver’s improvement of Lovász’s θ function bound for the maximum size of a clique in a graph.      

Zonal spherical functions on CROSS’s and special functions

We find formulas for the eigenvalues of the Laplacian and the zonal spherical functions on all simply-connected CROSS??s by a simple method, using the trigonometric formulas of spherical geometry, Hopf fiber bundles, and the results on the spectra of the Laplacian on the total space and on the base of a Riemannian submersion with totally geodesic fibers. We find direct relations of the so-obtained zonal spherical functions to the special functions: hypergeometric finite Gauss series, Jacobi polynomials, and orthogonal polynomials including the ultraspherical Gegenbauer polynomials whose particular cases are given by the Legendre polynomials and the Chebyshev polynomials of the first and second kinds. We point out the relations to the corresponding results by Helgason and Berger with coauthors and give brief information about the method of calculating the spectra of the Laplacian on compact simply-connected irreducible Riemannian spaces and the spectra of the Laplacian on the CROSS??s obtained therefrom.     

On global quadratic growth condition for min-max optimization problems with quadratic functions

Second-order sufficient condition and quadratic growth condition play important roles both in sensitivity and stability analysis and in numerical analysis for optimization problems. In this article, we concentrate on the global quadratic growth condition and study its relations with global second-order sufficient conditions for min-max optimization problems with quadratic functions. In general, the global second-order sufficient condition implies the global quadratic growth condition. In the case of two quadratic functions involved, we have the equivalence of the two conditions.     

Epi-convergence: The Moreau Envelope and Generalized Linear-Quadratic Functions

This work explores the class of generalized linear-quadratic functions, constructed using maximally monotone symmetric linear relations. Calculus rules and properties of the Moreau envelope for this class of functions are developed. In finite dimensions, on a metric space defined by Moreau envelopes, we consider the epigraphical limit of a sequence of quadratic functions and categorize the results. We examine the question of when a quadratic function is a Moreau envelope of a generalized linear-quadratic function; characterizations involving nonexpansiveness and Lipschitz continuity are established. This work generalizes some results by Hiriart-Urruty and by Rockafellar and Wets.     

UNIVERSAL FUNCTIONS IN PARTIAL STRUCTURES

In this work we show that every structure ?? can be expanded to a partial structure ??* with universal functions for the class of polynomials on ??*. We can embed ??* monomorphically in a total structure ??º that preserves universal functions of ??* and that is universal among such structures, i.e. ??º can be homomorphically embedded in every total structure that preserves universal functions of ??*. Universal functions are the starting point for developing recursion theoretic tools in an ??* that satisfies some simple additional conditions.     

On the analysis of a simple evolutionary algorithm on quadratic pseudo-boolean functions

Evolutionary algorithms are randomized search heuristics, which are often used as function optimizers. In this paper the well-known (1+1) Evolutionary Algorithm ((1+1) EA) and its multistart variants are studied. Several results on the expected runtime of the (1+1) EA on linear or unimodal functions have already been presented by other authors. This paper is focused on quadratic pseudo-boolean functions, i.e., polynomials of degree 2, a class of functions containing NP-hard optimization problems. Subclasses of the class of all quadratic functions are identified where the (1+1) EA is efficient, for other subclasses the (1+1) EA has exponential expected runtime, but a large enough success probability within polynomial time such that a multistart variant of the (1+1) EA is efficient. Finally, a particular quadratic function is identified where the EA and its multistart variants fail in polynomial time with overwhelming probability.     

A variational approach to optimal control problems for elastic body motions

The initial-boundary problem for the linear theory of elasticity is considered. Based on the method of integrodifferential relations a new dynamical variational principle in which displacement, stress, and momentum functions are varied is proposed and discussed. To minimize the nonnegative functional under initial, boundary, and partial differential constraints arising in this approach a regular algorithm for approximation of the unknown functions is worked out. The algorithm gives us the possibility to estimate explicitly the local and integral quality of obtained numerical solutions. An effective numerical method for the optimization problems of controlled motions of elastic bodies with quadratic objective functionals is developed. As example, the 3D problems of optimal longitudinal motions of a rectilinear elastic prism with a quadratic cross section are considered for the terminal total mechanical energy to be minimized. The numerical results and their error estimates are presented and discussed. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On convex relaxations for quadratically constrained quadratic programming

We consider convex relaxations for the problem of minimizing a (possibly nonconvex) quadratic objective subject to linear and (possibly nonconvex) quadratic constraints. Let $\mathcal{F }$ denote the feasible region for the linear constraints. We first show that replacing the quadratic objective and constraint functions with their convex lower envelopes on $\mathcal{F }$ is dominated by an alternative methodology based on convexifying the range of the quadratic form $\genfrac(){0.0pt}{}{1}{x}\genfrac(){0.0pt}{}{1}{x}^T$ for $x\in \mathcal{F }$ . We next show that the use of ?? $\alpha $ BB?? underestimators as computable estimates of convex lower envelopes is dominated by a relaxation of the convex hull of the quadratic form that imposes semidefiniteness and linear constraints on diagonal terms. Finally, we show that the use of a large class of D.C. (??difference of convex??) underestimators is dominated by a relaxation that combines semidefiniteness with RLT constraints.     

Undominated d.c. Decompositions of Quadratic Functions and Applications to Branch-and-Bound Approaches

In this paper we analyze difference-of-convex (d.c.) decompositions for indefinite quadratic functions. Given a quadratic function, there are many possible ways to decompose it as a difference of two convex quadratic functions. Some decompositions are dominated, in the sense that other decompositions exist with a lower curvature. Obviously, undominated decompositions are of particular interest. We provide three different characterizations of such decompositions, and show that there is an infinity of undominated decompositions for indefinite quadratic functions. Moreover, two different procedures will be suggested to find an undominated decomposition starting from a generic one. Finally, we address applications where undominated d.c.d.s may be helpful: in particular, we show how to improve bounds in branch-and-bound procedures for quadratic optimization problems.     

Remarks on the degree of instability

Linear systems of differential equations allowing of functions in quadratic forms that do not increase along trajectories with time are considered. The relations between the indices of inertia of these forms and the degrees of instability of equilibrium states are indicated. These assertions generalize known results from the oscillation theory of linear systems with dissipation, and reveal the mechanism of loss of stability when non-increasing quadratic forms lose the property of a minimum.     

Robust discrimination under a hierarchy on the scatter matrices

Under normality, Flury and Schmid [Quadratic discriminant functions with constraints on the covariances matrices: some asymptotic results, J. Multivariate Anal. 40 (1992) 244-261] investigated the asymptotic properties of the quadratic discrimination procedure under hierarchical models for the scatter matrices, that is: (i) arbitrary scatter matrices, (ii) common principal components, (iii) proportional scatter matrices and (iv) identical matrices. In this paper, we study the properties of robust quadratic discrimination rules based on robust estimates of the involved parameters. Our analysis is based on the partial influence functions of the functionals related to these parameters and allows to derive the asymptotic variances of the estimated coefficients under models (i)-(iv). From them, we conclude that the asymptotic variances verify the same order relations as those obtained by Flury and Schmid [Quadratic discriminant functions with constraints on the covariances matrices: some asymptotic results, J. Multivariate Anal. 40 (1992) 244-261] for the classical estimators. We also perform a Monte Carlo study for different sample sizes and different hierarchies which shows the advantage of using robust procedures over classical ones, when anomalous data are present. It also confirms that better rates of misclassification can be achieved if a more parsimonious model among all the correct ones is used instead of the standard quadratic discrimination.     

A STOCHASTIC FEEDBACK MODEL FOR OPTIMAL MANAGEMENT OF RENEWABLE RESOURCES

Analytical expressions for optimal harvest of a renewable resource stock which is subject to a stochastic process are found. These expressions give the optimal harvest as an explicit feedback control law. All relations in the model, including the stochastic process, may be arbitrary functions of the state variable (stock). The objective function, however, is at most a quadratic function in the control variable (yield). A quadratic objective function includes the cases of downward sloping demand and increasing marginal costs which are the most common sources for nonlinearities in the economic part of the model. When it is assumed that there is a moratorium on harvest for stock sizes below a certain level (biological barrier), it is shown that the barrier requirements influence the optimal harvest paths throughout.     

Real polynomial iterative roots in the case of nonmonotonicity height ≥ 2

It is known that a strictly piecewise monotone function with nonmonotonicity height ≥ 2 on a compact interval has no iterative roots of order greater than the number of forts. An open question is: Does it have iterative roots of order less than or equal to the number of forts? An answer was given recently in the case of "equal to". Since many theories of resultant and algebraic varieties can be applied to computation of polynomials, a special class of strictly piecewise monotone functions, in this paper we investigate the question in the case of "less than" for polynomials. For this purpose we extend the question from a compact interval to the whole real line and give a procedure of computation for real polynomial iterative roots. Applying the procedure together with the theory of discriminants, we find all real quartic polynomials of non-monotonicity height 2 which have quadratic polynomial iterative roots of order 2 and answer the question.     

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??This paper considers the expected penalty functions for a discrete semi-Markov risk model, which includes several existing risk models such as the compound binomial model (with time-correlated claims) and the compound Markov binomial model (with time-correlated claims) as special cases. Recursive formulae and the initial values for the discounted free penalty functions are derived in the two-state model by an easy method. We also give some applications of our results.     

Some Orthogonal Polynomials Related to Elliptic Functions

We characterize the orthogonal polynomials in a class of polynomials defined through their generating functions. This led to three new systems of orthogonal polynomials whose generating functions and orthogonality relations involve elliptic functions. The Hamburger moment problems associated with these polynomials are indeterminate. We give infinite families of weight functions in each case. The different polynomials treated in this work are also polynomials in a parameter and as functions of this parameter they are orthogonal with respect to unique measures, which we find explicitly. Through a quadratic transformation we find a new exactly solvable birth and death process with quartic birth and death rates.     

Mean-variance versus expected utility in dynamic investment analysis

Given the existence of a Markovian state price density process, this paper extends Merton??s continuous time (instantaneous) mean-variance analysis and the mutual fund separation theory in which the risky fund can be chosen to be the growth optimal portfolio. The CAPM obtains without the assumption of log-normality for prices. The optimal investment policies for the case of a hyperbolic absolute risk aversion (HARA) utility function are derived analytically. It is proved that only the quadratic utility exhibits the global mean-variance efficiency among the family of HARA utility functions. A numerical comparison is made between the growth optimal portfolio and the mean-variance analysis for the case of log-normal prices. The optimal choice of target return which maximizes the probability that the mean-variance analysis outperforms the expected utility portfolio is discussed. Mean variance analysis is better near the mean and the expected utility maximization is better in the tails.     

Quadratic transformations of the sixth Painlevé equation with application to algebraic solutions

In 1991, one of the authors showed the existence of quadratic transformations between the Painlevé VI equations with local monodromy differences (1/2, a, b, ±1/2) and (a, a, b, b). In the present paper we give concise forms of these transformations. They are related to the quadratic transformations obtained by Manin and Ramani–Grammaticos–Tamizhmani via Okamoto transformations. To avoid cumbersome expressions with differentiation, we use contiguous relations instead of the Okamoto transformations. The 1991 transformation is particularly important as it can be realized as a quadratic‐pull back transformation of isomonodromic Fuchsian equations. The new formulas are illustrated by derivation of explicit expressions for several complicated algebraic Painlevé VI functions. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the structure of convex piecewise quadratic functions

Convex piecewise quadratic functions (CPQF) play an important role in mathematical programming, and yet their structure has not been fully studied. In this paper, these functions are categorized into difference-definite and difference-indefinite types. We show that, for either type, the expressions of a CPQF on neighboring polyhedra in its domain can differ only by a quadratic function related to the common boundary of the polyhedra. Specifically, we prove that the monitoring function in extended linear-quadratic programming is difference-definite. We then study the case where the domain of the difference-definite CPQF is a union of boxes, which arises in many applications. We prove that any such function must be a sum of a convex quadratic function and a separable CPQF. Hence, their minimization problems can be reformulated as monotropic piecewise quadratic programs.This research was supported by Grant DDM-87-21709 of the National Science Foundation.     

Objects, Discreteness, and Pure Power Theories:

Sydney Shoemaker??s causal theory of properties is an important starting place for some contemporary metaphysical perspectives concerning the nature of properties. In this paper, I discuss the causal and intrinsic criteria that Shoemaker stipulates for the identity of genuine properties and relations, and address George Molnar??s criticism that holding both criteria presents an unbridgeable hypothesis in the causal theory of properties. The causal criterion requires that properties and relations contribute to the causal powers of objects if they are to be deemed genuine rather than ??mere-Cambridge??. The intrinsic criterion requires that all genuine properties and relations be intrinsic. Molnar??s S-property argument says that these criteria conflict if one considers extrinsic spatiotemporal properties and relations to contribute causally. In this paper, I argue that a solution to the contradiction that Molnar identifies involves a denial of discreteness between objects, leading to a power holist perspective and a resulting deflationary account of intrinsicality.