Semi-infinite Problem Research Articles

Robust Semi-Infinite Interval Equilibrium Problem Involving Data Uncertainty: Optimality Conditions and Duality

In this paper, we model uncertainty in both the objective function and the constraints for the robust semi-infinite interval equilibrium problem involving data uncertainty. We particularize these conditions for the robust semi-infinite mathematical programming problem with interval-valued functions by extending the results from the literature. We introduce the dual robust version of the above problem, prove the Mond–Weir-type weak and strong duality theorems, and illustrate our results with an example.

An Inexact Bundle Method for Semi-Infinite Minimax Problems

Semi-infinite minimax problems are widely utilized in various fields; however, there is a scarcity of algorithms that can directly tackle convex-convex and convex-concave semi-infinite minimax problems. An inexact algorithm based on the bundle method is introduced in this paper, which can be directly applied to solve both types of semi-infinite minimax problems. The novel algorithm offers the advantage of not requiring exact solutions for the inner maximization problem but only necessitates optimal solution with a certain level of precision. Additionally, the augmentation function method is employed to address nonconvergence issues encountered in traditional bundle method when dealing with convex-convex minimax problems. Global convergence of our algorithm is proven under reasonable assumptions. Numerical results from several examples demonstrate the effectiveness and practicality of our proposed approach.

On the convergence of local solutions for the method quADAPT

ABSTRACT A classical solution approach to semi-infinite programming, which is easy to implement, is based on discretizing the semi-infinite index set. The Blankenship and Falk algorithm adaptively chooses a small discretization. On every iteration, a solution based on the current discretization is calculated. In a second step, the most violated constraint is determined and added to the discretization. In a previous work, the authors showed that the algorithm has a slow convergence and introduced a new method that exhibits a quadratic rate [Seidel and Küfer, An adaptive discretization method solving semi-infinite optimization problems with quadratic rate of convergence. Optimization. 2022;71(8):2211–2239]. In this paper, we further investigate the introduced method. The method implements new constraints that can cut off parts of the feasible set. We will study the effect on local minima. We assume that in each iteration local solutions to the discretized problems are computed. We will give an example showing that in general a limit point is not necessarily a local solution of the original semi-infinite problem, but only of an approximate problem. We then study second-order conditions and show that they coincide for both problems. We use this to develop conditions under which local solutions converge to a local solution in the limit. Finally, we present quadratic convergence results for the case of local solutions.

Robust optimality conditions for semi-infinite equilibrium problems involving data uncertainty

Robust optimality conditions for semi-infinite equilibrium problems involving data uncertainty

On Semi-Infinite Optimization Problems with Vanishing Constraints Involving Interval-Valued Functions

In this paper, we examine a semi-infinite interval-valued optimization problem with vanishing constraints (SIVOPVC) that lacks differentiability and involves constraints that tend to vanish. We give definitions of generalized convex functions along with supportive examples. We investigate duality theorems for the SIVOPVC problem. We establish these theorems by creating duality models, which establish a relationship between SIVOPVC and its corresponding dual models, assuming generalized convexity conditions. Some examples are also given to illustrate the results.

A quantum algorithm for computing dispersal of submarine volcanic tephra

In this paper, we develop a quantum computing algorithm for solving the partial differential equation (PDE) for tephra dispersal through advection in the semi-infinite horizontal buoyant region of a submarine volcanic eruption. The concentration of pyroclastic particles in the fluid domain of a hydrothermal megaplume provides important information about the rate of volcanic energy release, mechanism of formation of the megaplume, and submarine depositional patterns. This work leveraging on previous works [F. Gaitan, NPJ Quantum Inf. 6, 61 (2020); F. Gaitan, Adv. Quantum Tech. 4, 2100055 (2021)] further opens up opportunities to solve wider classes of PDEs with different applications of interest. Some additional specific contributions of this work are transforming the semi-infinite spatial domain problem into a problem on a finite spatial domain for applying the quantum algorithm, and the investigation into the effect of spatial and temporal resolution on the solution of PDEs for the quantum algorithm. Furthermore, possible modification of the algorithm with different spatial discretization schemes has been presented and their influence and implications on the solution of the PDE have been discussed. Also, studies are conducted to examine the effect of regularity conditions in time and the presence of statistical noise in the spatial domain, on the solutions obtained using quantum algorithms. The study in this paper paves an important pathway to venture into other types of advection-diffusion problems.

Approximate proper efficiencies in nonsmooth semi-infinite multiobjective optimization problems

This article is devoted to studying a nonsmooth semiinfinite multiobjective optimization problem (SIMP) in terms of the Mordukhovich/limiting subdifferentials. We first establish necessary and sufficient conditions for an ε−quasi positively properly efficient solution of a problem (SIMP). We also investigate Mond-Weir type dual problems under assumptions of ε−quasi pseudo-generalized convexity. Next, we provide an application to a nonsmooth fractional semi-infinite multiobjective optimization problem. Finally, some examples are given to illustrate the obtained results. The obtained results improve or include some recent known ones.

Multi-Scale-Matching neural networks for thin plate bending problem

Physics-informed neural networks are a useful machine learning method for solving differential equations, but encounter challenges in effectively learning thin boundary layers within singular perturbation problems. To resolve this issue, multi-scale-matching neural networks are proposed to solve the singular perturbation problems. Inspired by matched asymptotic expansions, the solution is decomposed into inner solutions for small scales and outer solutions for large scales, corresponding to boundary layers and outer regions, respectively. Moreover, to conform neural networks, we introduce exponential stretched variables in the boundary layers to avoid semi-infinite region problems. Numerical results for the thin plate problem validate the proposed method.

Local isolated efficiency in non-smooth robust semi-infinite multi-objective fractional programming problems

Local isolated efficiency in non-smooth robust semi-infinite multi-objective fractional programming problems

On approximate positively properly efficient solutions in nonsmooth semi-infinite multiobjective optimization problems with data uncertainty

On approximate positively properly efficient solutions in nonsmooth semi-infinite multiobjective optimization problems with data uncertainty

Three-dimensional flow of hybrid nanofluid through Darcy-Forchheimer porous surface: A Legendre wavelet collocation approach

The main objective of the present study is to obtain the solution of 3D Darcy-Forchheimer flow of hybrid nanofluid (MoS2-GO/H2O) over stretchable surface using a novel Legendre Wavelet Collocation Method (LWCM). The transformed model is a semi-infinite boundary value problem of nonlinear coupled system of Ordinary Differential Equation (ODEs). Using special transformation in a space coordinate, the model is converted into finite domain of boundary value problem. Then, LWCM is implemented and problem is converted into system of algebraic equations. Now, the solution of algebraic equations is obtained by Newton-Raphson method. The ratio of volume fraction of GO and MoS2 are taken as 6% and 8% respectively. The outcomes disclose that the heat transfer rate and both drag forces regularly rises with increase in volume fraction of MoS2 and GO. Moreover, the heat transfer outlines of MoS2-GO/H2O constantly decreased when the fluid motion is in suction region, while the skin friction coefficients are increased. To validate the results obtained by the LWCM, a comparison is made with exact solution as well as with earlier published numerical results.

Set-based robust optimization of uncertain multiobjective problems via epigraphical reformulations

In this paper, we study a method for finding robust solutions to multiobjective optimization problems under uncertainty. We follow the set-based minmax approach for handling the uncertainties which leads to a certain set optimization problem with the strict upper type set relation. We introduce, under some assumptions, a reformulation using instead the strict lower type set relation without sacrificing the compactness property of the image sets. This allows to apply vectorization results to characterize the optimal solutions of these set optimization problems as optimal solutions of a multiobjective optimization problem. We end up with multiobjective semi-infinite problems which can then be studied with classical techniques from the literature.

Semi-infinite interval equilibrium problems: optimality conditions and existence results

This paper aims to obtain new Karush–Kuhn–Tucker optimality conditions for solutions to semi-infinite interval equilibrium problems with interval-valued objective functions. The Karush–Kuhn–Tucker conditions for the semi-infinite interval programming problem are particular cases of those found in this paper for constrained equilibrium problem. We illustrate this with some examples. In addition, we obtain solutions to the interval equilibrium problem in the unconstrained case. The results presented in this paper extend the corresponding results in the literature.

A hybrid artificial boundary combining perfectly matched layer and infinite elements for analysing semi-infinite problems

Artificial boundaries are crucial for simulating the dynamic response of semi-infinite problems. In this paper, a novel hybrid artificial boundary for the simulation of the far field is proposed by combining the perfectly matched layer (PML) and infinite element method (IEM), while the near field is simulated by the finite element method (FEM). The new hybrid artificial boundary (PML-IEM) is devised such that it can enjoy the relative advantage of each method. To start, a brief description is given on the fundamentals of the PML, IEM and the proposed PML-IEM. Then, the procedure for implementing the FEM-PML (using finite elements) and the IEM (using infinite elements) in the stretched coordinates is derived. In the parametric analysis, the factors affecting the accuracy of the PML, IEM, and the proposed PML-IEM are assessed in details. Based on this, optimal parameters are selected for implementing the proposed PML-IEM. Through the examples, the accuracy and reliability of the hybrid method proposed for analysing the semi-infinite problems under the surface line loads are demonstrated to be superior to those by the pure PML or IEM.

$$\varepsilon $$-Quasi-Weakly Solution for Semi-infinite Vector Optimization Problems with Data Uncertainty

$$\varepsilon $$-Quasi-Weakly Solution for Semi-infinite Vector Optimization Problems with Data Uncertainty

Robust Beamforming Design for RIS-Aided Integrated Sensing and Communication System

It is expected that the future intelligent transportation system will be endowed with the sensing ability to cope with the complex road environment. Therefore, the integrated sensing and communications (ISAC) system can complement the development of intelligent transportation. In this work, a novel reconfigurable intelligent surface (RIS)-aided ISAC system is investigated, in which an RIS reflects signals to the vehicle target and user by creating a directional path to enhance sensing and communication performance. We are interested in the joint robust design of transmitted beamformer at the dual-functional radar-communication (DFRC) base station and phase-shift at the RIS to maximize the radar mutual information subject to user achievable rate constraint under imperfect angles knowledge and channel state information (CSI). Specifically, two CSI error models, namely, the bounded and the mixed bounded-moment error models, are considered. Then, a worst-case robust (WCR) beamforming problem, as well as a mixed chance-constrained and worst-case robust (MCWR) beamforming problem, are separately formulated. Furthermore, we develop two efficient methods to convert the formulated semi-infinite constraint problems into feasibility ones, and an alternate optimization framework is proposed to obtain stationary points of the original problems. Simulation results are provided to validate the effectiveness of the proposed transformation methods and solution.

A Convex Formulation for Robust Control of Bilinear Positive Systems With Evolutionary Dynamics

In this article, optimal control of bilinear positive systems with uncertain evolutionary dynamics is studied. The first objective is to determine optimal control signals such that the effect of the disturbance signals on the outputs of the nominal dynamics is minimized in terms of the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal{H}_2$</tex-math> </inline-formula> norm performance index. Properties of the positive systems, Metzler matrices, and the exploitable structure of the problem are utilized to show that the underlying problem is representable as a standard convex optimization problem. Next, the optimal strategy is examined in the presence of polytopic uncertainties. In this case, the optimization problem is a semi-infinite min-max problem. It is shown that under some assumptions, the underlying problem is equivalent to a finite min-max problem, i.e., it suffices to check only the vertices of the polytopic set for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal{H}_2$</tex-math> </inline-formula> norm minimization. The proposed approach for polytopic bilinear positive systems is non-iterative, does not involve parameter tuning or nonsmooth optimization algorithms, and can be solved using off-the-shelf solvers. The effectiveness of the proposed approach is investigated for the optimal combination therapy of human immunodeficiency virus (HIV).

Nonsmooth semi-infinite roughly B-invex multi-objective programming problems

In this paper, we will study a nonsmooth semi-infinite multi-objective programming problem involving roughly B-invex functions. Sufficient optimality conditions for the primal problem are derived. We shall formulate Mond–Weir type dual and establish weak and strong duality theorems under various roughly B-invexity assumptions.

An SDP method for fractional semi-infinite programming problems with SOS-convex polynomials

In this paper, we study a class of fractional semi-infinite polynomial programming problems involving sos-convex polynomial functions. For such a problem, by a conic reformulation proposed in our previous work and the quadratic modules associated with the index set, a hierarchy of semidefinite programming (SDP) relaxations can be constructed and convergent upper bounds of the optimum can be obtained. In this paper, by introducing Lasserre’s measure-based representation of nonnegative polynomials on the index set to the conic reformulation, we present a new SDP relaxation method for the considered problem. This method enables us to compute convergent lower bounds of the optimum and extract approximate minimizers. Moreover, for a set defined by infinitely many sos-convex polynomial inequalities, we obtain a procedure to construct a convergent sequence of outer approximations which have semidefinite representations (SDr). The convergence rate of the lower bounds and outer SDr approximations are also discussed.

Semi-infinite Mathematical Programming Problems involving Generalized Convexity

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