Infinite Number Of Constraints Research Articles

Hadamard well-posedness for a set optimization problem with an infinite number of constraints

This article aims to study the Hadamard well-posedness for a set optimization problem with an infinite number of constraints, where Kuroiwa's lower set less relation is used to compare entire images of an objective set-valued map. Based on the weak Slater constraint qualification, sufficient conditions of Hadamard well-posedness properties for efficient and weakly efficient solutions to the reference problem under functional perturbations of both constraint sets and objective maps are established. The obtained results can also be seen as an extension to the case of infinite/semi-infinite vector optimization problems.

Kernel-based identification with frequency domain side-information

This paper discusses the problem of system identification when frequency domain side-information is available. We mainly consider the case where the side-information is provided as the H∞-norm of the system being bounded by a given scalar. This framework allows considering different forms of frequency domain side-information, such as the dissipativity of the system. We propose a nonparametric identification approach for estimating the impulse response of the system under the given side-information. The estimation problem is formulated as a constrained optimization in a stable reproducing kernel Hilbert space, where suitable constraints are considered for incorporating the desired frequency domain features. The resulting optimization has an infinite-dimensional feasible set with an infinite number of constraints. We show that this problem is a well-defined convex program with a unique solution. We propose a heuristic that tightly approximates this unique solution. The proposed approach is equivalent to solving a finite-dimensional convex quadratically constrained quadratic program. The efficiency of the discussed method is verified by several numerical examples.

Dynamic Pricing with Money‐Back Guarantees

Money‐back guarantees (MBGs), which allow customers to return products that do not meet their expectations, are widely used in the retail industry. In this study, we study a retailer's MBG policy with dynamic pricing of limited inventory. A key decision for the retailer is to decide whether to offer MBGs. When the product can be returned instantly, we find that the optimal MBG policy is a simple threshold policy: given the inventory level, it is optimal to offer an MBG if and only if the remaining selling time is longer than a threshold. Moreover, the threshold is decreasing with the inventory level. We also address the problem of dynamic pricing with positive return times. Due to the complexity, we analyze the associated fluid model, which has an infinite number of constraints. We consider a series of relaxations that have a nested structure and use the Lagrangian approach to explicitly solve these relaxed problems. This allows us to develop an iterative approach that is guaranteed to solve the fluid model in finite iterations. Our numerical analysis shows that the deterministic solution is asymptotically optimal for the stochastic system.

An Exact Algorithm Based on the Kuhn–Tucker Conditions for Solving Linear Generalized Semi‐Infinite Programming Problems

Optimization problems containing a finite number of variables and an infinite number of constraints are called semi‐infinite programming problems. Under certain conditions, a class of these problems can be represented as bi‐level programming problems. Bi‐level problems are a particular class of optimization problems, in which there is another optimization problem embedded in one of the constraints. We reformulate a semi‐infinite problem into a bi‐level problem and then into a single‐level nonlinear one by using the Kuhn–Tucker optimality conditions. The resulting reformulation allows us to employ a branch and bound scheme to optimally solve the problem. Computational experimentation over well‐known instances shows the effectiveness of the developed method concluding that it is able to effectively solve linear semi‐infinite programming problems. Additionally, some linear bi‐level problems from literature are used to validate the robustness of the proposed algorithm.

Optimality Conditions for Minimax Optimization Problems with an Infinite Number of Constraints and Related Applications

Optimality Conditions for Minimax Optimization Problems with an Infinite Number of Constraints and Related Applications

Robust Beamforming Design for Magnetic MIMO Wireless Power Transfer Systems

In this paper, we investigate the robust beamforming for the magnetic-based wireless power transfer (WPT) with multiple transmitters and multiple receivers to improve the power efficiency and reliability. By taking into account the imperfection of the magnetic mutual inductances information (MII), we firstly formulate a beamforming optimization problem to minimize the total transmit power while guaranteeing the reliable power delivery for each receiver simultaneously. Then, observing that the optimization problem is highly nonconvex with infinite number of constraints, we utilize the norm-bounded MII error models to reformulate the original problem into two worst-case optimization problems for the cases of the single receiver and multiple receivers, respectively. Finally, the optimal beamforming is obtained for the case of single receiver by applying the semidefinite programming relaxation (SDR) and the approximately optimal solution is obtained for the case of multiple receivers by adopting the SDR with randomization. Extensive simulation results verify the power transfer reliability and efficiency of the proposed beamforming schemes for the magnetic-based WPT systems.

A note on minimax optimization problems with an infinite number of constraints

A note on minimax optimization problems with an infinite number of constraints

Nonlinear System Identification With Prior Knowledge on the Region of Attraction

We consider the problem of nonlinear system identification when prior knowledge is available on the region of attraction (ROA) of an equilibrium point. We propose an identification method in the form of an optimization problem, minimizing the fitting error and guaranteeing the desired stability property. The problem is approached by joint identification of the dynamics and a Lyapunov function verifying the stability property. In this setting, the hypothesis set is a reproducing kernel Hilbert space, and with respect to each point of the given subset of the ROA, the Lie derivative inequality of the Lyapunov function imposes a constraint. The problem is a non-convex infinite-dimensional optimization with an infinite number of constraints. To obtain a tractable formulation, only a suitably designed finite subset of the constraints are considered. The resulting problem admits a solution in form of a linear combination of the sections of the kernel and its derivatives. An equivalent finite dimension optimization problem with a quadratic cost function subject to linear and bilinear constraints is derived. A suitable change of variable gives a convex reformulation of the problem. The method is demonstrated by several examples.

Efficient estimation of smoothing spline with exact shape constraints

ABSTRACT Smoothing spline is a popular method in non-parametric function estimation. For the analysis of data from real applications, specific shapes on the estimated function are often required to ensure the estimated function undeviating from the domain knowledge. In this work, we focus on constructing the exact shape constrained smoothing spline with efficient estimation. The ‘exact’ here is referred as to impose the shape constraint on an infinite set such as an interval in one-dimensional case. Thus the estimation becomes a so-called semi-infinite optimisation problem with an infinite number of constraints. The proposed method is able to establish a sufficient and necessary condition for transforming the exact shape constraints to a finite number of constraints, leading to efficient estimation of the shape constrained functions. The performance of the proposed methods is evaluated by both simulation and real case studies.

Optimality Conditions for Approximate Pareto Solutions of a Nonsmooth Vector Optimization Problem with an Infinite Number of Constraints

In this paper, we present some new necessary and sufficient optimality conditions in terms of Clarke subdifferentials for approximate Pareto solutions of a nonsmooth vector optimization problem which has an infinite number of constraints. As a consequence, we obtain optimality conditions for the particular cases of cone-constrained convex vector optimization problems and semidefinite vector optimization problems. Examples are given to illustrate the obtained results.

Weak-stability and saddle point theorems for a multiobjective optimization problem with an infinite number of constraints

Weak-stability and saddle point theorems for a multiobjective optimization problem with an infinite number of constraints

Dynamic Pricing with Money Back Guarantees

Money back guarantees (MBGs), which allow customers to return products that do not meet their expectations, are widely used in the retail industry. In this paper, we study a retailer's MBG policy with dynamic pricing of a limited inventory. A key decision for the retailer is to decide whether to offer MBGs. When the product can be returned instantly, we find that the optimal MBG is a simple threshold policy: given the inventory level, it is optimal to offer an MBG if and only if the remaining selling time is longer than a threshold. Moreover, the threshold is increasing in the inventory level. We also address the problem of dynamic pricing with positive return times. Due to its complexity, we analyze the associated fluid model, which has an infinite number of constraints. We consider a series of relaxations that have a nested structure and use the Lagrangian approach to explicitly solve these relaxed problems. This allows us to develop an iterative approach that is guaranteed to solve the fluid model in finite iterations. Our numerical analysis shows that the deterministic solution is asymptotically optimal for the stochastic system.

Robust Approximate Optimality Conditions for Uncertain Nonsmooth Optimization with Infinite Number of Constraints

This paper deals with robust quasi approximate optimal solutions for a nonsmooth semi-infinite optimization problems with uncertainty data. By virtue of the epigraphs of the conjugates of the constraint functions, we first introduce a robust type closed convex constraint qualification. Then, by using the robust type closed convex constraint qualification and robust optimization technique, we obtain some necessary and sufficient optimality conditions for robust quasi approximate optimal solution and exact optimal solution of this nonsmooth uncertain semi-infinite optimization problem. Moreover, the obtained results in this paper are applied to a nonsmooth uncertain optimization problem with cone constraints.

Parametric Optimization of Systems with Distributed Parameters in Problems with Mixed Constraints on the Final States of the Object of Control

A constructive technology of the solution of the parametrized problems of the programmed optimal control of systems with the distributed parameters under the conditions of different requirements for the permissible deviation of the resulting spatial distribution of the controlled value from the set magnitude in the uniform metric is proposed. The developed technique uses a special procedure of the one-criterion convolution of the considered constraints and the subsequent reduction to the typical form of the problems of mathematical programming on the extremum of a function of a finite number of variables with an infinite number of constraints (semi-infinite optimization problem), which is solved by the scheme of the previously developed alternance method. An example of optimization by the criteria of the speed and energy consumption of unsteady heat conduction processes with two different restrictions on the accuracy of approximation to the given temperature conditions, which is of independent interest, is given.

Demand-driven interprocedural analysis for map-based abstract domains

Many data flow analysis problems can be succinctly formalized using constraint systems. For interprocedural analysis, the system may contain an infinite number of constraints, but it can still be solved using a local solver that evaluates the constraints in a demand-driven fashion. In this paper, we use local solvers to develop a compositional framework for interprocedural on-demand static analysis. We can integrate any map-based abstract domain into our framework, such as a points-to analysis that maps pointers to their possible target addresses. Driven by the local solving algorithm that tracks the required dependencies, only those points-to sets that are of interest to the user are computed. The approach is applicable whenever the keys of the map are efficiently comparable and the domain operations are applied pointwise; we place no additional restrictions on the value domain nor on the transfer functions of the analysis beyond the standard termination requirements of the solvers.

Правило множителей Лагранжа для общей задачи на экстремум с бесконечным числом ограничений

Правило множителей Лагранжа для общей задачи на экстремум с бесконечным числом ограничений

Simplex Algorithm for Countable-State Discounted Markov Decision Processes

We consider discounted Markov decision processes (MDPs) with countably-infinite state spaces, finite action spaces, and unbounded rewards. Typical examples of such MDPs are inventory management and queueing control problems in which there is no specific limit on the size of inventory or queue. Existing solution methods obtain a sequence of policies that converges to optimality in value but may not improve monotonically, ie., a policy in the sequence may be worse than preceding policies. Our proposed approach considers countably-infinite linear programming (CILP) formulations of the MDPs (a CILP is defined as a linear program (LP) with countably-infinite numbers of variables and constraints). Under standard assumptions for analyzing MDPs with countably-infinite state spaces and unbounded rewards, we extend the major theoretical extreme point and duality results to the resulting CILPs. Under additional mild assumptions, which are satisfied by several applications of interest, we present a simplex-type algorithm that is implementable in the sense that each of its iterations requires only a finite amount of data and computation. We show that the algorithm finds a sequence of policies that improves monotonically and converges to optimality in value. Unlike existing simplex-type algorithms for CILPs, our proposed algorithm solves a class of CILPs in which each constraint may contain an infinite number of variables and each variable may appear in an infinite number of constraints. A numerical illustration for inventory management problems is also presented. The online appendix is available at https://doi.org/10.1287/opre.2017.1598 .

Modelling non-equilibrium thermodynamic systems from the speed-gradient principle.

The application of the speed-gradient (SG) principle to the non-equilibrium distribution systems far away from thermodynamic equilibrium is investigated. The options for applying the SG principle to describe the non-equilibrium transport processes in real-world environments are discussed. Investigation of a non-equilibrium system's evolution at different scale levels via the SG principle allows for a fresh look at the thermodynamics problems associated with the behaviour of the system entropy. Generalized dynamic equations for finite and infinite number of constraints are proposed. It is shown that the stationary solution to the equations, resulting from the SG principle, entirely coincides with the locally equilibrium distribution function obtained by Zubarev. A new approach to describe time evolution of systems far from equilibrium is proposed based on application of the SG principle at the intermediate scale level of the system's internal structure. The problem of the high-rate shear flow of viscous fluid near the rigid plane plate is discussed. It is shown that the SG principle allows closed mathematical models of non-equilibrium processes to be constructed.This article is part of the themed issue 'Horizons of cybernetical physics'.

Aspects of optimization with stochastic dominance

We consider stochastic optimization problems with integral stochastic order constraints. This problem class is characterized by an infinite number of constraints indexed by a function space of increasing concave utility functions. We are interested in effective numerical methods and a Lagrangian duality theory. First, we show how sample average approximation and linear programming can be combined to provide a computational scheme for this problem class. Then, we compute the Lagrangian dual problem to gain more insight into this problem class.

Robust Optimization of Index-2 Differential Algebraic Equations with Guaranteed Stability under Parametric Uncertainty: Application to a Reactor–Separator–Recycle Process

In this article, we propose a method for steady-state optimal design of index-2 differential algebraic systems under parametric uncertainty. We use the matrix pencil to evaluate directly the stability of index-2 differential algebraic equations and formulate stability constraints using the Routh–Hurwitz test. The underlying mathematical problem is difficult to solve because it involves infinite stability constraints. We developed an algorithm where an infinite number of constraints can be implemented as several relaxation problems that are solved iteratively. Additionally, the simulation result under parametric uncertainty is used to estimate the bound of the state perturbations rather than assumptions based on experience that may lead to overly conservative or not implementable designs. To illustrate the method, we apply it to a reactor–separator–recycle process and obtain the robustly stable design.