Semi-infinite Problem Research Articles

Heat-induced planar shock waves in supercritical fluids

We have investigated one-dimensional compression waves, produced by heat addition in quiescent and uniform initial conditions, in six different supercritical fluids, each taken in four states ranging from compressible pseudo-liquid fluid to ideal gas. Navier–Stokes simulations of a canonical semi-infinite domain flow problem, spanning five orders of magnitude of heating rate, are also carried out to support the theoretical analysis. Depending on the intensity of the Gaussian-shaped energy source, linear waves or shock waves due to nonlinear wave steepening are observed. A new reference heating rate parameter allows to collapse in the linear regime the whole dataset, together with the existing experimental data, thanks to its absorption of real-fluid effects. Moreover, the scaling strategy illustrates a clear separation between linear and nonlinear regimes for all fluids and conditions, offering motivation for the derivation of a unified fully predictive model for shock intensity. The latter is performed by extending the validity of previously obtained theoretical results in the nonlinear regime to supercritical fluids. Finally, thermal to mechanical power conversion efficiencies are shown to be proportional, in the linear regime, to the fluid’s Gruneisen parameter, which is the highest for compressible pseudo-liquid fluids, and maximum in the nonlinear regime for ideal gases.

Optimality and Duality for Multiobjective Semi-infinite Variational Problem Using Higher-Order B-type I Functions

The notion of higher-order B-type I functional is introduced in this paper. This notion is utilized to study optimality and duality for multiobjective semi-infinite variational problem in which the index set of inequality constraints is an infinite set. The concept of efficiency is used as a tool for optimization. Mond–Weir type of dual is proposed for which weak, strong, and strict converse duality theorems are proved to relate efficient solutions of primal and dual problems.

A standard branch-and-bound approach for nonlinear semi-infinite problems

This paper considers nonlinear semi-infinite problems, which contain at least one semi-infinite constraint (SIC). The standard branch-and-bound algorithm is adapted to such problems by extending usual upper and lower bounding techniques for nonlinear inequality constraints to SICs. This is achieved by defining the interval evaluation of parametrized functions and their generalized gradients, by also adapting numerical constraint programming techniques to quantified inequalities, and by introducing linear relaxations and restrictions for SICs. The overall efficiency of our algorithm is demonstrated on a standard set of benchmarks from the literature, in comparison with the best state of the art alternative.

Generalized problem of linear copositive programming

We consider a special class of optimization problems where the objective function is linear w.r.t. decision variable х and the constraints are linear w.r.t. х and quadratic w.r.t. index t defined in a given cone. The problems of this class can be considered as a generalization of semi-definite and copositive programming problems. For these problems, we formulate an equivalent semi-infinite problem and define a set of immobile indices that is either empty or a union of a finite number of convex bounded polyhedra. We have studied properties of the feasible sets of the problems under consideration and use them to obtain new efficient optimality conditions for generalized copositive problems. These conditions are CQ-free and have the form of criteria.

Further exploration of model truncation to extend the applicability of the finite element method to higher frequencies

In a previous presentation, the use of model truncation to extend the applicability of the finite element method to higher frequencies was explored. This talk furthers that exploration and focuses on the development of an easy-to-implement alternative to the perfectly matched layer for the truncation of semi-infinite problems with coupled structural and acoustic domains. Numerical results showing the efficacy of the proposed method are discussed and a method to bound the error that results from treating a large finite structural as semi-infinite is considered. It is hoped that the proposed method helps facilitate extension of the finite element method to mid-frequency applications. [Work supported by ONR.]

On Semi-infinite Mathematical Programming Problems with Equilibrium Constraints Using Generalized Convexity

In this paper, we consider semi-infinite mathematical programming problems with equilibrium constraints (SIMPPEC). By using the notion of convexificators, we establish sufficient optimality conditions for the SIMPPEC. We formulate Wolfe and Mond–Weir-type dual models for the SIMPPEC under the invexity and generalized invexity assumptions. Weak and strong duality theorems are established to relate the SIMPPEC and two dual programs in the framework of convexificators.

Effect of the notch-component relative size on the notch fatigue limit of AISI 304L specimens under push–pull tests

Results of a large set of fatigue tests on AISI 304L stainless steel cylindrical round specimens with a circumferential notch of semicircular profile and subjected to uniaxial loading ( R = –1) are reported. The outer diameter of the specimens has been kept fixed, whereas the radius of the notch has been increased progressively. A whole range of configurations is studied, from combinations of notch radius and specimen diameter where the remaining ligament is much larger than the notch (semi-infinite problem) to configurations where the ligament is much smaller than the notch (finite problem) and where interaction effects have a noticeable influence on the shape of the stress gradient. An S–N curve has been obtained for each configuration. The experimentally obtained fatigue limits for the different notch radii have been compared with predictions made with several methods. However, a strong divergence between the experimental values and the theoretical predictions, which were extremely conservative, has been found and this has been attributed to work-hardening and residual stress effects connected with the machining process. It has also been suggested that a geometrical notch strengthening effect, possibly connected with stress triaxiality due to the notch, and the accompanying hydrostatic tensile stress, may also be a contributing factor. A simple practical engineering method to account for these effects has been proposed.

Nonsmooth Optimization Method for H∞ Output Feedback Control

This paper proposes a nonsmooth optimization method for [Formula: see text] output feedback control problem of linear time-invariant(LTI) systems based on bundle technique. We formulate this problem as a nonconvex and nonsmooth semi-infinite constrained optimization problem by quantifying both internal stability of closed-loop system and measurement of system performance, where [Formula: see text] norm of closed-loop transfer function and a stabilization channel is used. Our method uses progress function and bundle technique to solve the resulting problem which has a composite structure. We prove the convergence to a critical point from a feasible initial point and test some benchmarks to demonstrate the effectiveness of this method.

Optimal Robust Ordering Quantity for a New Product Under Environmental Constraints

Rapid new product introduction is an effective method for firms to capture market share. Considering the growing concerns of environmental threats, firms are under intense pressure to measure their impacts on the environment. Before launching a new product, a firm should make the appropriate decision regarding the order quantity with incomplete information to achieve the goals of both profit maximization and environmental threat minimization simultaneously. In this paper, distribution-free news vendor models with the environmental constraints are proposed to obtain efficient order decisions. The models are formulated based on the absolute minimax regret criterion. Based on the moment bound problem framework, the models are translated into a semi-infinite linear optimization problem. In addition, the closed-form solutions of the robust order quantities and carbon emission volumes are derived. The characteristics of effective order decisions are discussed under three types of carbon emissions regulations (i.e., the carbon capacity policy, carbon tax scheme, and carbon trading mechanism). By analyzing the product characteristics from both economic and environmental aspects, four possible new product introduction strategies are proposed that the firm might employ under different environmental constraints. The characteristics of the optimal order quantity are also investigated under three carbon emissions regulations across four new product introduction strategies.

Stability in unified semi-infinite vector optimization

This paper is devoted to the study of Painleve−Kuratowski convergence of constraint sets, minimal and efficient solution sets of a unified semi-infinite vector optimization problem. The convergence is established under the functional perturbations of objective function and constraint maps assuming the continuous convergence of the objective functions and uniform gamma convergence of the constraint maps. Further, examples are formulated to analyze the significance of assumptions in the main results.

Necessary optimality conditions for a nonsmooth semi-infinite programming problem

The nonsmooth semi-infinite programming $$\left( {\textit{SIP}}\right) $$ is solved in the paper (Mishra et al. in J Glob Optim 53:285–296, 2012) using limiting subdifferentials. The necessary optimality condition obtained by the authors, as well as its proof, is false. Even in the case where the index set is a finite, the result remains false. Two major problems do not allow them to have the expected result; first, the authors were based on Theorem 3.2 (Soleimani-damaneh and Jahanshahloo in J Math Anal Appl 328:281–286, 2007) which is not valid for nonsmooth semi-infinite problems with an infinite index set; second, they would have had to assume a suitable constraint qualification to get the expected necessary optimality conditions. For the convenience of the reader, under a nonsmooth limiting constraint qualification, using techniques from variational analysis, we propose another proof to detect necessary optimality conditions in terms of Karush–Kuhn–Tucker multipliers. The obtained results are formulated using limiting subdifferentials and Frechet subdifferentials.

Lagrange duality and saddle point optimality conditions for semi-infinite mathematical programming problems with equilibrium constraints

In this paper, we consider a special class of optimization problems which contains infinitely many inequality constraints and finitely many complementarity constraints known as the semi-infinite mathematical programming problem with equilibrium constraints (SIMPEC). We propose Lagrange type dual model for the SIMPEC and obtain their duality results using convexity assumptions. Further, we discuss the saddle point optimality conditions for the SIMPEC. Some examples are given to illustrate the obtained results.

A Sum-Of-Squares Constrained Regression Approach for Process Modeling

Abstract Combining empirical relationships with a backbone of first-principle laws allow the modeler to transfer the available process knowledge into a model. In order to get such so-called grey-box models, data-reconciliation methods and constrained regression algorithms are key to obtain reliable process models that will be used later for optimization. However, the existent approaches require solving a semi-infinite constrained regression nonlinear problem, which is usually done numerically by an iterative procedure alternating between a relaxed problem and an a posteriori check for constraint violation. This paper proposes an alternative one-stage efficient approach for polynomial regression models based in sum-of-squares (convex) programming. Moreover, it is shown how several desirable features on the regression model can be naturally enforced in this optimization framework. The effectiveness of the proposed ap-proach is illustrated through an academic example provided in the related literature.

ϵ-Efficient solutions in semi-infinite multiobjective optimization

In this paper we apply some tools of nonsmooth analysis and scalarization method due to Chankong–Haimes to find ϵ-efficient solutions of semi-infinite multiobjective optimization problems (MP). We establish ϵ-optimality conditions of Karush–Kuhn–Tucker (KKT) type under Farkas–Minkowski (FM) constraint qualification by using ϵ-subdifferential concept. In addition we propose mixed type dual problem (including dual problems of Wolfe and Mond–Weir types as special cases) for ϵ-efficient solutions and investigate relationship between mentioned (MP) and its dual problem as well as establish several ϵ-duality theorems.

An Approach to $\epsilon$-duality Theorems for Nonconvex Semi-infinite Multiobjective Optimization Problems

Using a scheme for solving multiobjective optimization problems via a system of corresponding scalar problems, approximate optimality conditions for a nonconvex semi-infinite multiobjective optimization problem are established. As a new approach, the scheme is developed to study the approximate duality theorems of the problem via a pair of primal-dual scalar problems. Several $\epsilon$-duality theorems are given. Furthermore, the existence theorem for almost quasi weakly $\epsilon$-Pareto solutions of the primal problem, and the existence theorem for quasi weakly $\epsilon$-Pareto solutions of the dual problem are established without any constraint qualification.

Elasto-plastic solution for shallow tunnel in semi-infinite space

• Elasto-plastic solution for shallow tunnel in a semi-infinite space was proposed. • The gravitational effect on stress in the elastic surrounding rocks is notable. • The influence of gravity on the elasto-plastic model is limited. • Calculations of plastic zone shape are consistent with the results generated by Midas-GTS. This study focuses on a novel approach to develop an elasto-plastic solution for surrounding rocks around a shallow tunnel in an elasto-plastic semi-infinite space that incorporates the gravitational effect based on the bipolar coordinate system. A shallow circular tunnel was excavated in continuous, isotropic, and homogeneous rock masses. The solution for the surrounding rock in semi-infinite space was regarded as a semi-infinite plane problem, and the semi-infinite plane was partitioned along the curve trajectories of the bipolar coordinate. Based on the Mohr-Coulomb failure criterion and considering the gravitational effect, the closed-form solutions for the stress distribution of the surrounding rock were proposed. Furthermore, novel approaches for the calculation of the shape of the plastic zone around the tunnel and the critical internal support pressure were obtained. These presented approaches were validated by comparing the results of finite element method and the published literature. Using the numerical method, the sensitivity of gravity and the influences of the geological strength index, internal support pressure, surface loading, and buried depth of the tunnel on the plastic radius, shape of the plastic zone, and stress distribution of the surrounding rocks around the shallow circular tunnel were discussed.

On First-Order Conditions for Optimality of Nondifferentiable Semi-infinite Programming

The purpose of this paper is to give some new Karush–Kuhn–Tucker-type necessary optimality conditions for nonsmooth semi-infinite problems. Moreover, we present some suitable examples for our results. The paper is organized by Frechet and Mordukhovich subdifferentials.

Filter Trust Region Method for Nonlinear Semi-Infinite Programming Problem

We present a filter trust region method for nonlinear semi-infinite programming. Based on the discretization technique and motivated by the multiobjective programming, we transform the semi-infinite problem into a finite one. Together with the filter technique, we propose a modified method that avoids the merit function. Compared with the existing methods, our method is more flexible and easier to implement. Under some mild conditions, the convergent properties are proved. Moreover, the numerical results are reported in the end.

Control of fracture at the interface of dissimilar materials using randomly oriented inclusions and networks

Fracture at or near the interface between two isotropic materials has been a subject of extensive research relevant to the problems encountered in aerospace, naval, electronic packaging, biomechanical engineering and other applications. The problems of edge cracks, semi-infinite interface cracks and substrate cracks under a thermally stressed film considered in the paper are representative of such analyses. We consider a possible improvement in the fracture resistance achieved by embedding randomly distributed stiff inclusions such as fibers, nanotubes, fiber or nanotube networks and ellipsoidal or spherical particles in the more compliant material. This results in a smaller mismatch between the stiffness of the joined materials that may prevent or alleviate fracture. Numerical examples demonstrate both the benefits and the limitations of enhancing the stiffness of the compliant material. In particular, the examples refer to the boundary between singular and non-singular interfacial edge stresses attempting to avoid the stress singularity by embedding random carbon nanotubes or networks. In the semi-infinite interfacial crack problem it is demonstrated that a small reduction in the stiffness mismatch of two materials at the interface causes a significant decrease in the strain energy release rate. The approach to the analysis of substrate cracks under a thermally stressed film is outlined accounting for the history of thermal loading and the effect of temperature on the properties of the film, substrate and embedded inclusions. The solutions presented in the paper rely on effective properties of randomly reinforced materials. The limitations of such approach in fracture problems and an estimate of its validity based on a comparison of scales at the tip of the crack and in the representative volume cell are discussed. It is suggested that fracture involving nanoparticle and nanotube reinforced materials can be characterized using effective properties.

Comparison of analytical solution of the semi-infinite problem of soil freezing with numerical solutions in various simulation software

The objective of this research is to determine the most suitable software packages for simulation of thermal effect of various constructions on seasonally frozen or permafrost ground. The abrupt change in soil physical properties at freeze/thaw boundaries is a considerable challenge because it introduces a strong nonlinearity into the heat transfer model. Mathematically speaking, there appears a Stefan condition at the moving boundary of the phase transition. A problem of soil column freezing is considered. The simulation results, obtained by different software, are compared with the analytical solution of the corresponding semi-infinite Stefan problem. The frost penetration depth and the depth profiles of temperature and its absolute error, after 300 days, are presented.