Semi-infinite Problem Research Articles

Optimality and duality for nonsmooth semi-infinite multiobjective programming with support functions

In this paper, we consider a nonsmooth semi-infinite multiobjective programming problem involving support functions. We establish sufficient optimality conditions for the primal problem. We formulate Mond-Weir type dual for the primal problem and establish weak, strong and strict converse duality theorems under various generalized convexity assumptions. Moreover, some special cases of our problem and results are presented.

Electroelastic Green’s function of one-dimensional piezoelectric quasicrystals subjected to multi-physics loads

Green’s functions of infinite and semi-infinite plane problems of one-dimensional quasicrystals with piezoelectric effect are obtained in a closed form by Stroh formalism. Some numerical examples under different loading conditions, such as line forces, line dislocations, and a line charge, are given to explain the mechanical and electric behaviors of the quasicrystals. Various elastic–electric constants of quasicrystals are analyzed and the coupling effects between the phonon and phason fields are studied. The presented solutions will be useful for many boundary value problems of one-dimensional quasicrystals with piezoelectric effect. Moreover, the numerical results can be used to verify the accuracy of the solutions by some numerical methods, such as the finite element and boundary element methods. Furthermore, the Stroh formalism can be generalized to the researches on more complex problems of quasicrystals.

On optimality conditions and duality theorems for robust semi-infinite multiobjective optimization problems

In this paper, we consider a semi-infinite multiobjective optimization problem with more than two differentiable objective functions and uncertain constraint functions, which is called a robust semi-infinite multiobjective optimization problem and give its robust counterpart \({\mathrm{(RSIMP)}}\) of the problem, which is regarded as the worst case of the uncertain semi-infinite multiobjective optimization problem. We prove a necessary optimality theorem for a weakly robust efficient solution of \({\mathrm{(RSIMP)}} \), and then give a sufficient optimality theorem for a weakly robust efficient solution of \({\mathrm{(RSIMP)}}\). We formulate a Wolfe type dual problem of \({\mathrm{(RSIMP)}}\) and give duality results which hold between \({\mathrm{(RSIMP)}}\) and its dual problem.

Normal regularity for the feasible set of semi-infinite multiobjective optimization problems with applications

We establish verifiable conditions for the feasible set of a nonsmooth semi-infinite multiobjective optimization problem to have the normal regularity (that is, the coincidence of the Frechet normal cone and the limiting normal one) at a given point. In this way, both the Frechet normal cone and the limiting normal one to the considered set are then computed via active constraint multipliers and limiting subdifferentials of the involved constraints. In order to achieve such goals, two classes of nonsmooth functions are introduced and exploited. Finally, the obtained results are applied to provide necessary optimality conditions for semi-infinite multiobjective optimization problems.

Stability of Efficient Solutions for Semi-infinite Vector Optimization Problems

Stability of Efficient Solutions for Semi-infinite Vector Optimization Problems

An Efficient Algorithm for Min-Max Convex Semi-Infinite Programming Problems

ABSTRACTIn this article, we consider the convex min-max problem with infinite constraints. We propose an exchange method to solve the problem by using efficient inactive constraint dropping rules. There is no need to solve the maximization problem over the metric space, as the algorithm has merely to find some points in the metric space such that a certain criterion is satisfied at each iteration. Under some mild assumptions, the proposed algorithm is shown to terminate in a finite number of iterations and to provide an approximate solution to the original problem. Preliminary numerical results with the algorithm are promising. To our knowledge, this article is the first one conceived to apply explicit exchange methods for solving nonlinear semi-infinite convex min-max problems.

Method of Outer Approximations and Adaptive Approximations for a Class of Matrix Games

We present a novel technique for obtaining global solutions to discrete min-max problems that arise naturally in the receding horizon control of unmanned craft in which the controls can be adjusted only in notches, e.g., stop, half forward, full forward, left or right $$60^{\circ }$$60?, and in the finite precision global solution of certain classes of semi-infinite min-max problems. The technique consists of a method for transcribing min-max problems over discrete sets into a matrix game and matrix game-specific adaptations of the Method of Outer Approximations and the Method of Adaptive Approximations, which are normally used for solving optimal control and semi-infinite min-max problems. The efficiency of the Method of Outer Approximations depends on having a good initial approximation to a solution. To this end, we make use of adaptive approximation techniques to decompose a large matrix game into a sequence of lower dimensional games, the solution of each giving rise to a very good initial approximation to a solution for the next game. We show that a basic approach for solving a min-max matrix game, in which one maximizes over the elements of columns and minimizes over the elements of the rows of a matrix, requires a number of function evaluations which is equal to the product of the number of rows and the number of columns of the matrix, while with our new approach our experimental results require only a number of function evaluations which is the product of the number rows and a number ranging from 2 to 20, shortening computing times from years to fractions of a minute.

Algorithms for the estimation of transient surface heat flux during ultra-fast surface cooling

Surface heat flux is an important parameter in various industrial applications. This paper compared several algorithms for estimating surface heat flux using both simulated heat flux and measured temperature in response to ultra-fast surface cooling. Two different strategies were employed to measure the surface temperature, using the thin film thermocouple with a 2μm depth deposited directly on the cooling substrate surface (direct surface temperature measurement) and the fine thermocouple with a 100μm bead diameter placed on the surface of the cooling substrate and covered with aluminum foil (indirect surface temperature measurement). The algorithms of Duhamel’s theorem, sequential function specification method (SFSM) and transfer function method are briefly analyzed. A new method is proposed that the surface temperature is first calculated based on Duhamel’s theorem and then the surface heat flux can be estimated, in order to improve the accuracy of results in the case of the indirect surface temperature measurement method. The transfer function is obtained by solving an auxiliary problem and thus the transfer function method can be implemented to solve multilayer, semi-infinite inverse heat conduction problems. Accuracy and sensitivity to noise are examined using both the simulated triangular pulse heat flux and the measured temperature data. Duhamel’s theorem is insensitive to noise, but is unsuitable for predicting surface heat flux in the indirect measurement method of surface temperature. The SFS method can provide acceptable results of surface heat flux using both measurement methods. However, a noticeable discrepancy exists as the heat flux changes extremely quickly and it suffers considerably from noise. The transfer function and newly proposed methods are both effective in inhibiting noise, and produce very similar results, which match the simulated heat flux exactly and have a negligible standard deviation and residual temperature.

A probabilistic interpretation of set-membership filtering: Application to polynomial systems through polytopic bounding

Set-membership estimation is usually formulated in the context of set-valued calculus and no probabilistic calculations are necessary. In this paper, we show that set-membership estimation can be equivalently formulated in the probabilistic setting by employing sets of probability measures. Inference in set-membership estimation is thus carried out by computing expectations with respect to the updated set of probability measures P as in the probabilistic case. In particular, it is shown that inference can be performed by solving a particular semi-infinite linear programming problem, which is a special case of the truncated moment problem in which only the zeroth order moment is known (i.e., the support). By writing the dual of the above semi-infinite linear programming problem, it is shown that, if the nonlinearities in the measurement and process equations are polynomial and if the bounding sets for initial state, process and measurement noises are described by polynomial inequalities, then an approximation of this semi-infinite linear programming problem can efficiently be obtained by using the theory of sum-of-squares polynomial optimization. We then derive a smart greedy procedure to compute a polytopic outer-approximation of the true membership-set, by computing the minimum-volume polytope that outer-bounds the set that includes all the means computed with respect to P.

Integral identities based on symmetric and skew-symmetric weight functions for a semi-infinite interfacial crack in anisotropic magnetoelectroelastic bimaterials

In this paper, we address a semi-infinite interfacial crack problem in an anisotropic magnetoelectroelastic (MEE) bimaterial system subjected to a magnetoelectromechanical asymmetric load on the crack surface. First, the symmetric and skew-symmetric weight functions are derived for a two-dimensional (2-D) deformation problem. Using these weight functions and extending the Betti formula to MEE materials, the integral identities are further obtained and the present crack problem is formulated in terms of singular integral equations, which establish the relationship between the applied external load and the generalized displacement jump across the crack faces. The illustrative examples in relation to Mode III, and Mode I and Mode II problems show that the method developed in this study avoids the use of Green's function and is very convenient for the fracture analysis of MEE solids, in which a multi-field coupled effect is observed.

A comparative note on the relaxation algorithms for the linear semi-infinite feasibility problem

The problem (LFP) of finding a feasible solution to a given linear semi-infinite system arises in different contexts. This paper provides an empirical comparative study of relaxation algorithms for (LFP). In this study we consider, together with the classical algorithm, implemented with different values of the fixed parameter (the step size), a new relaxation algorithm with random parameter which outperforms the classical one in most test problems whatever fixed parameter is taken. This new algorithm converges geometrically to a feasible solution under mild conditions. The relaxation algorithms under comparison have been implemented using the extended cutting angle method for solving the global optimization subproblems.

Necessary Optimality Conditions for Multi-Objective Semi-Infinite Variational Problem

In this paper, necessary optimality conditions for a class of Semi-infinite Variational Problems are established which are further generalized to a class of Multi-objective Semi-Infinite Variational Problems. These conditions are responsible for the development of duality theory which is an extremely important feature for any class of problems, but the literature available so far lacks these necessary optimality conditions for the stated problem. A lemma is also proved to find the topological dual of as it is required to prove the desired result.

尤拉方程在半无界区域的边值问题的基本解

本文把永久美式期权确定最佳实施边界的问题归结为尤拉方程在半无界区域的边值问题的基本解来研究。获得了基本解的表达式，同时确定了基本解的奇异点。证明了基本解在半无界区域连续，但解的导数在奇异点发生间断，在奇异点处基本解取最大值。基本解的奇异点就是永久美式期权最佳实施边界点。 In this paper, the permanent American options determine optimal exercise boundary problem is boiling down to the Euler equation in a semi infinite domain boundary value problem of the basic solution to study. We obtained the expression of the basic solution and the singular point of the basic solution. It is proved that the basic solution is continuous in the semi infinite domain, but the derivative of the solution is discontinuous at the singular point. The maximum value is obtained in the singular point. The singular point of the basic solution is the best implementation point of the permanent American option.

Sufficient optimality conditions for semi-infinite multiobjective fractional programming under (Ф,ρ)-V-invexity and generalized (Ф,ρ)-V-invexity

A new class of nonconvex smooth semi-infinite multiobjective fractional programming problems with both inequality and equality constraints is considered. We formulate and establish several parametric sufficient optimality conditions for efficient solutions in such nonconvex vector optimization problems under (?,?)-V-invexity and/or generalized (?,?)-V-invexity hypotheses. With the reference to the said functions, we extend some results of efficiency for a larger class of nonconvex smooth semi-infinite multiobjective programming problems in comparison to those ones previously established in the literature under other generalized convexity notions. Namely, we prove the sufficient optimality conditions for such nonconvex semi-infinite multiobjective fractional programming problems in which not all functions constituting them have the fundamental property of convexity, invexity and most generalized convexity notions.

Linearized Impulsive Fixed-Time Fuel-Optimal Space rendezvous: A New Numerical Approach

This paper focuses on the fixed-time minimum-fuel rendezvous between close elliptic orbits of an active spacecraft with a passive target spacecraft, assuming a linear impulsive setting and a Keplerian relative motion. Following earlier works developed in the 1960s, the original optimal control problem is transformed into a semi-infinite convex optimization problem using a relaxation scheme and duality theory in normed linear spaces. A new numerical convergent algorithm based on discretization methods is designed to solve this problem. Its solution is then used in a general simple procedure dedicated to the computation of the optimal velocity increments and optimal impulses locations. It is also shown that the semi-infinite convex programming has an analytical solution for the out-of-plane rendezvous problem. Different realistic numerical examples illustrate these results.

On strong KKT type sufficient optimality conditions for nonsmooth multiobjective semi-infinite mathematical programming problems with equilibrium constraints

In this paper, we consider a nonsmooth multiobjective semi-infinite mathematical programming problems with equilibrium constraints (MOSIMPECs). We introduce the concept of Mordukhovich stationary point for the nonsmooth multiobjective semi-infinite mathematical programming problems with equilibrium constraints in terms of the Clarke subdifferentials. Further, we establish that the M-stationary conditions introduced in this paper are strong KKT type sufficient optimality conditions for the nonsmooth multiobjective semi-infinite mathematical programming problems with equilibrium constraints under generalized invexity assumptions. We also illustrate our result with an example.

Constrained incremental bundle method with partial inexact oracle for nonsmooth convex semi-infinite programming problems

Semi-infinite problem (SIPs) are widely used in many control systems for solving complex control problem, such as polymerase chain reaction control system or other real time control system. In this paper, we present a bundle method for solving the nonsmooth convex SIPs, with the aim of working on the basis of "improvement function", "inexact oracle" and "incomplete knowledge" of the constraints. The proposed algorithm, whenever a new stabilized center is refreshed, requires an evaluation within some accuracy for the value of constraints. Beyond that, by using the incremental technique, it does not require all information about the constraints, but only one component function value and one subgradient needed to be estimated to update the bundle information and generate the search direction. Thus the computational cost is significantly reduced. Global convergence of this method is established based on some mild assumptions. Numerical experiments show that the algorithm is efficient for solving nonsmooth convex SIPs.

The sufficient efficiency conditions in semiinfinite multiobjective fractional programming under higher order exponential type hybrid type invexities

First, a class of higher order exponential type hybrid (α, β, γ, η, ρ, h(·, ·), κ(·, ·), Ω(·, ·, ·), ϖ(·, ·, ·), θ)-invexities is introduced, second, some parametrically sufficient efficiency conditions based on the higher order exponential type hybrid invexities are established, and finally some parametrically sufficient efficiency results under the higher order exponential type hybrid (α, β, γ, η, ρ, h(·, ·), κ(·, ·), Ω(·, ·, ·), ϖ(·, ·, ·), θ)-invexities are investigated to the context of solving semiinfinite multiobjective fractional programming problems. The notions of the higher order exponential type hybrid (α, β, γ, η, ρ, h(·, ·), κ(·, ·), Ω(·, ·, ·), ϖ(·, ·, ·), θ)-invexities encompass most of the generalized invexities in the literature. To the best of our knowledge, the results on semiinfinite multiobjective fractional programming problems established in this communication are new and application-oriented toward multitime multiobjectve problems as well as multiobjective control problems.

Model formulas for facilitating determination of concentration-dependent diffusion coefficients

Several formulas for determining the concentration dependence of diffusion coefficients are introduced for a one-dimensional semi-infinite diffusion problem, applying the Boltzmann-Matano method to “S-shaped” concentration profiles approximated by model functions. The functions are expressed in terms of the Gauss error function, hyperbolic tangent, exponential, and inverse tangent. For all model profiles the corresponding formulas for the diffusion coefficient are calculated. Rapid estimates of the diffusion coefficient are also provided as simple expressions obtained by evaluating the formulas at the center of the concentration profile. The results for the individual profiles are compared, and it is demonstrated that even very similar profiles can lead to rather different diffusion coefficients, especially at low concentrations. Using two examples of different diffusion processes, it is demonstrated that the results can be employed to rapidly calculate diffusion coefficients. In addition, it is shown that a finite diffusion coefficient at low concentrations only occurs if the corresponding concentration profile decays at a Gaussian rate or faster.

Slater CQ, optimality and duality for quasiconvex semi-infinite optimization problems

In this paper, we introduce and study the Slater constraint qualification (CQ) for a semi-infinite optimization problem with upper-semicontinuous quasiconvex objective and constraint functions. Then, some Karush–Kuhn–Tucker (KKT) type necessary and sufficient optimality conditions as well as duality results are derived. The final part of the paper is devoted to a linear characterization of optimality and the gap function for considered semi-infinite problem.