Constrained Maximization Problem Research Articles

Long Time Behavior of the Continuum Limit¶of the Toda Lattice, and the Generation¶of Infinitely Many Gaps from C ∞ Initial Data

We analyze a continuum limit of the finite non-periodic Toda lattice through an associated constrained maximization problem over spectral density functions. The maximization problem was derived by Deift and McLaughlin using the Lax–Levermore approach, initially developed for the zero dispersion limit of the KdV equation. It encodes the evolution of the continuum limit for all times, including evolution through shocks. The formation of gaps in the support of the maximizer is indicative of oscillations in the Toda lattice and the lack of strong convergence of the continuum limit. For large times, the maximizer tends to have zero gaps, which is the continuum analogue of the sorting property of the finite lattice. Using methods from logarithmic potential theory, we show that this behavior depends crucially on the initial data. We exhibit initial data for which the zero gap ansatz holds uniformly in the spatial parameter (at large times), and other initial data for which this uniformity fails at all times. We then construct an example of C ∞ smooth initial data generating, at a later time, infinitely many gaps in the support of the maximizer, while for even larger times, all gaps have closed.

On the supremum in linear fractional programming with respect to any closed unbounded feasible set

We consider a constrained maximization problem with a linear fractional function f over any closed and unbounded set X. Starting the analysis from the behavior of unbounded sequences in X, we find necessary and sufficient conditions for the supremum to be finite. We point out that when f does not have maximum value, the supremum is not necessarily attained along a recession direction. It is known that this fundamental property holds in the classical fractional problem with a polyhedral feasible region and we give a new proof of this result by using our approach and the Representation Theorem.

Genetic approach to the design of associative memory

In this paper, a genetic learning algorithm for neuron-weighted associative memory is presented. According to the criterions that measures the stability and attraction of associative memory, we cast the learning procedure into a global constrained maximization problem, solved by a genetic algorithm. This learning rule guarantees the storage of training patterns with basins of attraction as large as possible, which is realized by an appropriately defined discrete evaluation index. A larger attraction basin implies higher noise correction ability of associative memory. However, automatic adjustment of the attraction basin is difficult to realize by other method. To evaluate the performance of our learning strategy, a large number of simulation have been carried out

The Median Voter according to GARP

This paper adapts the generalized axiom of revealed preference (GARP) empirical method to the public goods problem to test whether observed municipal public spending can be explained "as if" the city governments maximize the utility of the median income voter. It applies the test procedure for medium-size municipal governments in five Midwest states. The data are consistent with GARP and reveal that the local governments in the sample behave as if they maximize median voter utility once we control for the state-specific effects, government management structure, and population density. The median voter hypothesis is the proposition that local governments behave "as if" they maximize the utility of the median income voter in the jurisdiction. Although an important tool in public economic analysis, there has been no conclusive direct test of the maximization hypothesis to date. Revealed preference theory shows that any finite set of price and quantity observations satisfying the generalized axiom of revealed preference (GARP) can be rationalized by the constrained maximization of an increasing, continuous, concave utility function (Afriat 1967, 1973; Varian 1982). This paper adapts the revealed preference method to the public goods problem in order to test municipal spending data for consistency under GARP, thereby providing the first direct test of whether or not the local governments behave "as if" they maximize median voter utility. The median voter hypothesis represents a simple tool that is the most widely used characterization of local government behavior. Given its prominence in the literature, it is not surprising that the hypothesis itself has become the subject of scrutiny. The framework assumes direct or representative democracy without strategic behavior, in which competition among politicians ensures that the public sector bureaucracy responds efficiently to voters' desires. Public sector decision making reduces to an "as if" constrained maximization problem, a simplified picture of the outcome of collective action. Still, much of the empirical testing literature is motivated by concerns over whether the model is too simplistic. Nonetheless, it can be argued that, given its depiction as an "as if" outcome, the median voter hypothesis should be judged not by its descriptive accuracy of the public sector decision process but instead by how well it explains observed local fiscal choices. On this basis, the existing empirical evidence, although mixed, reveals a surprising amount of indirect support for the framework. One approach taken in the literature is to evaluate the median voter hypothesis by testing the predictive power of regression equations with median income and tax price terms for explaining state or local fiscal behavior. In this line of the

Learning automata in feedforward connectionist systems

This paper analyses the behaviour of a general class of learning automata algorithms for feedforward connectionist systems in an associative reinforcement learning environment. The type of connectionist system considered is also fairly general. The associative reinforcement learning task is first posed as a constrained maximization problem. The algorithm is approximated by an ordinary differential equation using weak convergence techniques. The equilibrium points of the ordinary differential equation are then compared with the solutions to the constrained maximization problem to show that the algorithm does behave as desired

A critical assessment of the political preference function approach in agricultural economics

The political preference function (PPF) approach has been used often in the agricultural economics literature in attempts to explain the causes of farm policy. The fundamental assumption of PPF studies is that government selects levels of policy instruments such that it maximizes a function of interest groups' welfare measures for the purpose of acquiring maximum political support. PPF studies often use the familiar neoclassical constrained maximization problem to measure relative political powers of interest groups involved in policy determination. Some weaknesses of the PPF approach have been discussed in the literature (Gardner, 1989; Alston and Carter, 1991; and von CramonTaubadel (VCT), 1992 **). While each of these critiques provides valuable insight into PPF methodology and its limitations, none of them clearly explains the simple, fundamental structure of the PPF framework. This neglect of certain fundamentals has led to confusion in the litera-

Peak Directivity Optimization Under Side Lobe Level Constraints in Antenna Arrays

ABSTRACT This paper deals with a technique for the optimization of peak directivity under side lobe level constraints. The antennas treated are rectangular grid arrays of isotropic sources. The directivity of such an array may be expressed as a ratio of two functionals, called the Rayleigh quotient. By means of the projection matrix, the constrained maximization problem is transformed into the problem of an unconstrained maximization of a normalized quadratic function. A closed-form solution for the optimum peak directivity and the corresponding current distribution are found by solving for the biggest eigenvalue and the corresponding eigenvector in the generalized eigenvalue problem. Two types of arrays are considered. First, for linear arrays, the peak, directivity in the main beam is optimized under side lobe level constraints. Second, for square arrays with a rectangular grid, with the array pattern generated via the Baklanov transformation, a technique for optimizing the peak directivity under desire...

The impact of changes in relative weights on the optimal solution of a maximization problem

This paper studies the impact of changes in relative weights on the solution of a constrained maximization problem which arises in a number of different economic situations. Conditions are obtained under which a given change in relative weights necessarily leads to certain patterns of changes in the optimal solution. An example is provided for illustration.

On syndicate sharing rules for unanimous project rankings

This paper extends the notion of individual certainty equivalent to group certainty equivalent, in order to derive sharing rules that ensure unanimity on project rankings within a syndicate. A constrained maximization problem is set up with optimal project-specific sharing rules as the solution. It is shown that the group certainty equivalent construct allows for experimental derivation of a syndicate's preference function from a procedure parallel to that presently employed for deriving preference functions of individuals. The analysis potentially has a role in examining the nature of optimal compensation systems (sharing rules) for closely held (non-traded) corporations and partnerships.

Analysis and design of circular plates using the workhardening adaptation criterion

A unified approach to both analysis (maximum load multiplier) and optimal design problems for workhardening adaptation of axialsimmetric plates is presented, via a continuous approach. The loads are assumed to arbitrarily vary but in a quasi-static manner and remaining inside a given convex loading domain. In such a framework, the assessment of the plastic deformations developed during the actual (a priori unknown) loading history is crucial, since these deformations may result to be excessive with respect to some given ductility criteria, eventhough workhardening adaptation has occured. The bounds on plastic deformations are formulated by means of a quadratic but convex function depending on some fictitious plastic strain. The nonlinear mathematical programming problem so obtained is linearized by a suitable decomposition of the nonlinear constraint into two equivalent constraints. The analysis problem is identified as a linear constrained maximization problem. The optimal design problem in the presence of limited ductility constraints, is identified as a nonlinear constrained minimization problem, the nonlinearity being due to both the objective function and to one only of the constraints: the latter however are herein shown to be suitably linearized. The optimality conditions of the mathematical programming problem are also studied and discussed, and the theory is applied to a case-study structure.

Sensitivity analysis in economics

The paper studies the local dependence on a parameter of a unique solution to a general inequality-equality constrained nonlinear programming problem. A theorem due to Fiacco is extended to deal with the case where the objective and constraint functions are defined only over the nonnegative orthant. Other results are developed which extend sensitivity (or comparative statics) theorems due to Samuelson and Silberberg for equality constrained maximization problems to general nonlinear programs.

Generalized exterior boundary-value problems and optimization for the Helmholtz equation

A class of radiation problems is considered for the Helmholtz equation in exterior domains bounded by a smooth surface on which Dirichlet, Neumann, or Robin boundary conditions are imposed. The problem of finding the boundary data which maximizes far field power in a restricted subset of far field directions is formulated as a constrained maximization problem. Existence of an optimal solution in a variety of control domains is established. The particular case when the boundary is circular and the control domain is the unit ball inL2 is treated in detail. An algorithm for constructing the optimal solution is derived and used to obtain explicit numerical results.

Optimal Life Histories, Optimal Notation, and the Value of Reproductive Value

Reproductive value is the central construct in life history optimization. In the optimal life history, the reproductive value at every age represents the solution to a constrained maximization problem. This holds true regardless of trade-offs across age classes. The recent literature arguing to the contrary has misunderstood the principle and has neglected the constraints in the maximization. In reviewing the matter, this paper considers maximization over segments of the life history, maximization over segments taken in reverse sequence, maximization of terms of a decomposition into instantaneous contributions at a given age, and maximization of reproductive value over alternative states such as the sexes. To clarify the relation between the maximization procedure and the constraints, a simple iterative approach to achieving maximization and simultaneously satisfying the constraints is described. To arrive at the instantaneous decomposition, it is shown that reproductive value arises as an adjoint variabl...

Maximum power signal restoration

A new method of signal restoration is presented for applications where the signal is a sparse, positive series of delta functions. The method is defined by solving the constrained maximization problem maximize <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f^{T}f</tex> subject to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f \geq 0</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\parallelg - Hf\parallel^{2} = \paralleln\parallel^{2}</tex> . An algorithm to obtain the solution to this problem is given. Results are shown that demonstrate the successful application of this method.

Selecting efficient forecasts

The purpose of this enquiry into the nature of efficient forecasts is to identify the choice process utilised by the analyst and/or forecasting agency when selecting one particular forecast rather than any other. Two properties of a forecast, reliability and precision, are treated as joint products of r and d expenditures. Rational planners will select forecasts which maximise their preferences subject to the r and d budget. Thus, the selection of an optimum forecast can be treated as a constrained maximisation problem.

Absolutely continuous constrained maximizers

The modern theory of differentiable demand functions is generalized to maximizers in a constrained maximization problem. Sufficient conditions are given for almost everywhere differentiability of maximizers in the constraining capacities or other parameters. Also, sufficient conditions are given for the very useful condition (N). Finally, the somewhat stronger pointwise Lipschitz property is shown. Applications analogous to those in economic theory are indicated.

Afriat and Revealed Preference Theory

Suppose that we can observe a number of decisions xi (where xi is a non-negative N dimensional vector for i = 1, 2, ..., I) which some decision-making unit has made and let us further suppose that each vector of decisions xi satisfies a linear constraint of the form pTxi < 1 for i = 1, 2, ..., I where pi is a given N dimensional vector which has positive components.3 Given the above framework, we may ask the following question: is the observed set of decisions {xi} consistent with the hypothesis that the decision-maker chose decision xi (for i = 1, 2, ..., I) because it maximized a real valued function of N variables, ., subject to the constraint pTx < 1? A related question is how may we use the observed data {pi; xi} i = 1, 2, ..., I in order to construct an approximation to the decision-maker's true 0, assuming that such a b exists. In Section 3 below, we will give an answer to the above two questions by using the observed data {pi; xi} to construct the coefficients of a linear programming problem [4]. If this linear programme has a positive solution, then it turns out that the observed data does not satisfy the hypothesis of consistency. If on the other hand, the objective function of the linear programme has a solution equal to zero, then we may use the solution to the linear programme to construct a real valued function 0 such that for i = 1, 2, ..., I the vector xi is a solution to the following constrained maximization problem:

The potential method for conditional maxima in the locally compact metric spaces

In the paper there is presented a method of reducing the constrained maximization problem to an unconditional one. The method consists of introducing a function, called potential, which at an unconditional maximum converges to the constrained maximum when a positive parameter of the potential tends to zero. A proof of such convergence for the locally compact metric spaces is given.

Duality for non-linear programming in aBanach space

A constrained maximization problem in a realBanach space is considered, where the objective function is a non-linear pseudo concave functional and constraints are given bym non linear quasi convex functionals. The optimality conditions and converse duality theorem, given byRitter, have been extended to this class of programming problems.

Some duality relationships for the generalized Neyman-Pearson problem

In 1963, Kuhn presented a dual problem to a relatively well-known location problem, variously referred to as the generalized Fermat problem and the Steiner-Weber problem. The purpose of this paper is to point out how Kuhn's results can be adapted to provide a dual to the generalized Neyman-Pearson problem, a problem of fundamental interest in statistics, which has applications in control theory and a number of other areas. The Neyman-Pearson problem, termed the dual problem, is a constrained maximization problem and may be considered to be a calculus-of-variations analog to the bounded-variable problem of linear programming. When the dual problem has equality constraints, the primal problem is an unconstrained minimization problem. Duality results are also obtained for the case where the dual problem has inequality constraints.