Infinite-dimensional Optimization Problem Research Articles

A penalty method for a finite-dimensional obstacle problem with derivative constraints

We propose a power penalty method for an obstacle problem arising from the discretization of an infinite-dimensional optimization problem involving differential operators in both its objective function and constraints. In this method we approximate the mixed nonlinear complementarity problem (NCP) arising from the KKT conditions of the discretized problem by a nonlinear penalty equation. We then show the solution to the penalty equation converges exponentially to that of the mixed NCP. Numerical results will be presented to demonstrate the theoretical convergence rates of the method.

Robust Scenario Formulations for Strategic Supply Chain Optimization under Uncertainty

Strategic supply chain optimization (SCO) problems are often modeled as two-stage optimization problems, in which the first-stage variables represent decisions on the development of the supply chain and the second-stage variables represent decisions on the operations of the supply chain. When uncertainty is explicitly considered, the problem becomes an intractable infinite-dimensional optimization problem, which is usually solved approximately using a scenario or a robust approach. This article proposes a novel synergy of the scenario and robust approaches for strategic SCO under uncertainty. Two formulations are developed, namely, naive robust scenario formulation and affinely adjustable robust scenario formulation. It is shown that both formulations can be reformulated into tractable deterministic optimization problems if the uncertainty is bounded by the infinity norm and the uncertain equality constraints can be reformulated into deterministic constraints without any assumption about the uncertainty r...

Dubins Traveling Salesman Problem with Neighborhoods: A Graph-Based Approach

We study the problem of finding the minimum-length curvature constrained closed path through a set of regions in the plane. This problem is referred to as the Dubins Traveling Salesperson Problem with Neighborhoods (DTSPN). An algorithm is presented that uses sampling to cast this infinite dimensional combinatorial optimization problem as a Generalized Traveling Salesperson Problem (GTSP) with intersecting node sets. The GTSP is then converted to an Asymmetric Traveling Salesperson Problem (ATSP) through a series of graph transformations, thus allowing the use of existing approximation algorithms. This algorithm is shown to perform no worse than the best existing DTSPN algorithm and is shown to perform significantly better when the regions overlap. We report on the application of this algorithm to route an Unmanned Aerial Vehicle (UAV) equipped with a radio to collect data from sparsely deployed ground sensors in a field demonstration of autonomous detection, localization, and verification of multiple acoustic events.

Wing Kinematics Optimization for Hovering Micro Air Vehicles Using Calculus of Variation

The weight and power constraints imposed on flapping-wing micro air vehicles necessitate optimal design of the flapping kinematics. To date, the approach adopted for kinematics optimization has been to assume specific functions for the Euler angles describing the wing motion with respect to the body. Then, optimization is performed on the parameters of these functions. In another approach, a number of instants over the flapping cycle are selected, and optimization is performed on the magnitude of the Euler angles at these instants. This latter approach provides more freedom for the variations of the Euler angles rather than confining them to certain patterns. Yet, in both approaches, finite-dimensional optimization is adopted and, as such, additional constraints are imposed. Considering that the problem is an infinite-dimensional optimization problem, we use in this work the calculus of variations to obtain true optimality. The combination of the quasi-steady aerodynamics and the calculus of variations approach yields an upper bound for the flapping-wing aerodynamic performance. This bound can be used as a basis for evaluating the performance of any realistic design by assessing the degree of closeness between that design and the true optimal performance.

On Directional Metric Subregularity and Second-Order Optimality Conditions for a Class of Nonsmooth Mathematical Programs

We study infinite dimensional optimization problems where the constraint mapping is given as the sum of a smooth function and a generalized polyhedral multifunction, e.g., the normal cone mapping of a convex polyhedral set. By using advanced techniques of variational analysis we obtain first-order and second-order characterizations, both necessary and sufficient, for directional metric subregularity of the constraint mapping. These results are used to obtain second-order optimality conditions for the optimization problem.

Unconstrained model predictive control and suboptimality estimates for nonlinear continuous-time systems

This paper presents a continuous-time version of recent results on unconstrained nonlinear model predictive control (MPC) schemes. Based on a controllability assumption and a corresponding infinite-dimensional optimization problem, performance estimates and stability conditions are derived in terms of the prediction horizon and the sampling time of the MPC controller. Moreover, improved estimates for small sampling times are discussed and a comparison to the application of the discrete-time results in a sampled-data context is provided.

An extension of the Basic Constraint Qualification to nonconvex vector optimization problems

In this paper a Basic Constraint Qualification is introduced for a nonconvex infinite-dimensional vector optimization problem extending the usual one from convex programming assuming the Hadamard differentiability of the maps. Corresponding KKT conditions are established by considering a decoupling of the constraint cone into half-spaces. This extension leads to generalized KKT conditions which are finer than the usual abstract multiplier rule. A second constraint qualification expressed directly in terms of the data is also introduced, which allows us to compute the contingent cone to the feasible set and, as a consequence, it is proven that this condition is a particular case of the first one. Relationship with other constraint qualifications in infinite-dimensional vector optimization, specially with the Kurcyuscz-Robinson-Zowe constraint qualification, are also given.

Feedback optimal control of distributed parameter systems by using finite-dimensional approximation schemes.

Optimal control for systems described by partial differential equations is investigated by proposing a methodology to design feedback controllers in approximate form. The approximation stems from constraining the control law to take on a fixed structure, where a finite number of free parameters can be suitably chosen. The original infinite-dimensional optimization problem is then reduced to a mathematical programming one of finite dimension that consists in optimizing the parameters. The solution of such a problem is performed by using sequential quadratic programming. Linear combinations of fixed and parameterized basis functions are used as the structure for the control law, thus giving rise to two different finite-dimensional approximation schemes. The proposed paradigm is general since it allows one to treat problems with distributed and boundary controls within the same approximation framework. It can be applied to systems described by either linear or nonlinear elliptic, parabolic, and hyperbolic equations in arbitrary multidimensional domains. Simulation results obtained in two case studies show the potentials of the proposed approach as compared with dynamic programming.

Variational Approach to Second-Order Optimality Conditions for Control Problems with Pure State Constraints

For optimal control problems with set-valued control constraints and pure state constraints we propose new second-order necessary optimality conditions. In addition to the usual second-order derivative of the Hamiltonian, these conditions contain extra terms involving second-order tangents to the set of feasible trajectory-control pairs at the extremal process under consideration. The second-order necessary optimality conditions of the present work are obtained by using a variational approach. In particular, we present a new second-order variational equation. This approach allows us to make direct proofs as opposed to the classical way of obtaining second-order necessary conditions by using an abstract infinite dimensional optimization problem. No convexity assumptions on the constraints are imposed and optimal controls are required to be merely measurable.

Contamination Source Determination in Water Distribution Networks

We consider a water distribution network where at a finite number of nodes, contaminant injection can occur. We consider the problem of the identification of the contaminations. This problem can be considered as an optimal control problem with a networked system that is governed by a transport reaction equation. The identification is based upon observations from a finite number of sensors. The corresponding infinite-dimensional optimization problem is defined in a Hilbert space setting. In order to guarantee that our optimization problem has a unique solution, a quadratic regularization term is added in the objective function. Under certain assumptions on the relations of the travel times through the pipes we obtain a representation of this optimization problem that allows the computation of the solution on a discrete time grid by solving finite-dimensional linear least squares problems. On these time grids, there is no discretization error since our approach is based upon an exact representation of the system state. This is useful to minimize potential impacts of contamination emergencies on consumers by helping to select locations to flush the contaminant out of the distribution network.

Fuzzy Raw Materials Allocation Problem

Abstract Based on credibility theory, this paper presents a new raw materials allocation (RMA) model in which the demands of products and the prices of surplus products are supposed to fuzzy variables, the involved restrictions mainly include the inventory of each raw material and the production capacity of company. Since the fuzzy variables in the RMA model are usually continuous variables and the exact analytical expression of objective function cannot be obtained, traditional optimization algorithms cannot be used to solve it. Therefore, we firstly apply an approximation approach (AA) to turn the original infinite-dimensional optimization problem into an approximating finite-dimensional one, then design a hybrid algorithm which combines neural network (NN) and particle swarm optimization (PSO) to solve the approximating model. A given numerical example illustrates that the designed algorithm works well.

Subspace-Modeling-Based Nonlinear Measurement for Process Design

In industry, many nonlinear processes can be approximated well by a linear model under a suitable design parameter, upon which a linear controller can effectively control these processes. The key problem is how to find this suitable design parameter, which is never considered in process design. In this paper, a novel design for control approach is proposed to design the process to have a satisfactory linear approximation. First, a subspace-modeling-based nonlinear measurement is proposed to avoid the infinite dimensional optimization problem in the traditional nonlinearity measurement. Then a particle-swarm-optimization-based design approach is developed to obtain the optimal design parameter through solving the nonconvex and nondifferential measurement problem. Finally, the proposed design is applied to design a practical curing oven and compared with the existing design.

Transportation Chance Constrained Model with Fuzzy Parameters

This paper presents how credibility theory and chance constrained optimization method can be efficiently applied for modelling and solving transportation problem in fuzzy environment. Since the proposed transportation model includes fuzzy variable coefficients defined through possibility distributions with infinite supports, it is infinite-dimensional optimization problem. Therefore, we can not solve directly it by conventional optimization algorithms. To overcome this difficulty, we will discuss the approximation of the fuzzy transportation chance constrained problem in this paper, and design a heuristic algorithm, which combines approximation method (AM), neural network (NN) and genetic algorithm (GA) algorithm to solve this transportation chance constrained model. Finally, we present one numerical example to show the feasibility and effectiveness of the proposed method.

Two-stage fuzzy production planning expected value model and its approximation method

This work develops a novel two-stage fuzzy optimization method for solving the multi-product multi-period (MPMP) production planning problem, in which the market demands and some of the inventory costs are assumed to be uncertainty and characterized by fuzzy variables with known possibility distributions. Some basic properties about the MPMP production planning problem are discussed. Since the fuzzy market demands and inventory costs usually have infinite supports, the proposed two-stage fuzzy MPMP production planning problem is an infinite-dimensional optimization problem that cannot be solved directly by conventional numerical solution methods. To overcome this difficulty, this paper adopts an approximation method (AM) to turn the original two-stage fuzzy MPMP production planning problem into a finite-dimensional optimization problem. The convergence about the AM is discussed to ensure the solution quality. After that, we design a heuristic algorithm, which combines the AM and simulated annealing (SA) algorithm, to solve the proposed two-stage fuzzy MPMP production planning problem. Finally, one real case study about a furniture manufacturing company is presented to illustrate the effectiveness and feasibility of the proposed modeling idea and designed algorithm.

Infimal convolution, c-subdifferentiability, and Fenchel duality in evenly convex optimization

In this paper we deal with strong Fenchel duality for infinite-dimensional optimization problems where both feasible set and objective function are evenly convex. To this aim, via perturbation approach, a conjugation scheme for evenly convex functions, based on generalized convex conjugation, is used. The key is to extend some well-known results from convex analysis, involving the sum of the epigraphs of two conjugate functions, the infimal convolution and the sum formula of e-subdifferentials for lower semicontinuous convex functions, to this more general framework.

Unconstrained Nonlinear Model Predictive Control and Suboptimality Estimates for Continuous-Time Systems

AbstractThis paper presents a continuous-time version of recent results on unconstrained nonlinear model predictive control (MPC) schemes. Based on a controllability assumption and a corresponding infinite-dimensional optimization problem, performance estimates and stability conditions are derived in terms of the prediction horizon and the sampling time of the MPC controller. Moreover, improved estimates for small sampling times are discussed and a comparison to the application of the discrete-time results in a sampled-data context is provided.

Multidimensional Mechanism Design: Finite-Dimensional Approximations and Efficient Computation

Multidimensional mechanism design problems have proven difficult to solve by extending techniques from the one-dimensional case. This paper considers mechanism design problems with multidimensional types when the seller's cost function is not separable across buyers. By adapting results obtained by Border [Border, K. 1991. Implementation of reduced form auctions: A geometric approach. Econometrica 59 1175–1187], we transform the seller's problem into a representation that only involves “interim” variables and eliminates the dimensionality dependence on the number of buyers. We show that the associated infinite-dimensional optimization problem posed by the theoretical model can be approximated arbitrarily well by a sequence of finite-dimensional linear programming problems. We provide an efficient—i.e., terminating in polynomial time in the problem size—method to compute the separation oracle associated with the Border constraints and incentive compatibility constraints. This implies that our finite-dimensional approximation is solvable in polynomial time. Finally, we illustrate how the numerical solutions of the finite-dimensional approximations can provide insights into the nature of optimal solutions to the infinite-dimensional problem in particular cases.

Admittance Identification from Point-wise Sound Pressure Measurements Using Reduced-order Modelling

In this work an acoustic application is studied. The goal is to estimate the complex-valued admittance from given point measurements of the sound pressure. This parameter identification problem is formulated in terms of an infinite-dimensional optimization problem. First- and second-order optimality conditions are discussed. For the numerical realization a reduced-order model based on proper orthogonal decomposition is used. Numerical examples illustrate the efficiency of the proposed approach.

Galerkin methods in dynamic stochastic programming

The Galerkin method is a classical technique for approximating infinite-dimensional optimization problems with finite-dimensional ones. When applied to convex multistage stochastic programmes, it yields computationally attractive alternatives to scenario tree-based discretizations. We describe its implementations for dynamic portfolio optimization problems and report some encouraging numerical results.

Discrete mechanics and optimal control: An analysis

The optimal control of a mechanical system is of crucial importance in many application areas. Typical examples are the determination of a time-minimal path in vehicle dynamics, a minimal energy trajectory in space mission design, or optimal motion sequences in robotics and biomechanics. In most cases, some sort of discretization of the original, infinite-dimensional optimization problem has to be performed in order to make the problem amenable to computations. The approach proposed in this paper is to directly discretize the variational description of the system's motion. The resulting optimization algorithm lets the discrete solution directly inherit characteristic structural properties from the continuous one like symmetries and integrals of the motion. We show that the DMOC (Discrete Mechanics and Optimal Control) approach is equivalent to a finite difference discretization of Hamilton's equations by a symplectic partitioned Runge-Kutta scheme and employ this fact in order to give a proof of convergence. The numerical performance of DMOC and its relationship to other existing optimal control methods are investigated.