Finite-dimensional Optimization Problem Research Articles

Hydraulic resistance coefficient identification in pipelines

We consider an approach to identifying the hydraulic resistance coefficient for a segment of a magistral pipe in transporting hydrocarbon raw material. The considered identification problem reduces to a class of parametric optimal control problems, and we propose to use efficient numerical methods developed for first order finite-dimensional optimization problems to solve it. To this purpose, we derive formulas for components of the objective functional's gradient in the space of the identified parameters. The resulting values for the vector being optimized can then be used to construct the identified function from some class of functions with interpolation and approximation methods. We also show the results of numerical experiments.

Bilevel Optimal Control With Final-State-Dependent Finite-Dimensional Lower Level

In this paper we discuss special bilevel optimal control problems where the upper level problem is an optimal control problem of ODEs with control and terminal constraints and the lower level problem is a finite-dimensional parametric optimization problem where the parameter is the final state of the state variable of the upper level. We tackle this problem using tools from nonsmooth analysis, optimization in Banach spaces, and bilevel programming to derive necessary optimality conditions of linearized Pontryagin-type.

Bilevel Optimal Control Problems with Pure State Constraints and Finite-dimensional Lower Level

This paper focuses on the development of optimality conditions for a bilevel optimal control problem with pure state constraints in the upper level and a finite-dimensional parametric optimization problem in the lower level. After transforming the problem into an equivalent single-level problem, we concentrate on the derivation of a necessary optimality condition of Pontryagin type. We point out some major difficulties arising from the bilevel structure of the original problem and its pure state constraints in the upper level leading to a degenerated maximum principle in the absence of constraint qualifications. Hence, we use a partial penalization approach and a well-known regularity condition for optimal control problems with pure state constraints to ensure the nondegeneracy of the derived maximum principle. Finally, we illustrate the applicability of the derived theory by means of a small example.

A NUMERICAL METHOD FOR SOLVING LINEAR–QUADRATIC CONTROL PROBLEMS WITH CONSTRAINTS

The paper is devoted to the optimal control problem for a linear system with integrally constrained control function. We study the problem of minimization of a linear terminal cost with terminal constraints given by a set of linear inequalities. For the solution of this problem we propose two-stage numerical algorithm, which is based on construction of the reachable set of the system. At the first stage we find a solution to finite–dimensional optimization problem with a linear objective function and linear and quadratic constraints. At the second stage we solve a standard linear–quadratic control problem, which admits a simple and effective solution.

Contingent claim pricing through a continuous time variational bargaining scheme

We consider a variational problem modelling the evolution with time of two probability measures representing the subjective beliefs of a couple of agents engaged in a continuous-time bargaining pricing scheme with the goal of finding a unique price for a contingent claim in a continuous-time financial market. This optimization problem is coupled with two finite dimensional portfolio optimization problems, one for each agent involved in the bargaining scheme. Under mild conditions, we prove that the optimization problem under consideration here admits a unique solution, yielding a unique price for the contingent claim.

An Open-loop criterion for the solvability of a closed-loop guidance problem with incomplete information: Linear control systems

The method of open-loop control packages is a tool for stating the solvability of guaranteed closed-loop control problems under incomplete information on the observed states. In this paper, a solution method is specified for the problem of guaranteed closed-loop guidance of a linear control system to a convex target set at a prescribed point in time. It is assumed that the observed signal on the system’s states is linear and the set of its admissible initial states is finite. It is proved that the problem under consideration is equivalent to the problem of open-loop guidance of an extended linear control system to an extended convex target set. Using a separation theorem for convex sets, we derive a solvability criterion, which reduces to solving a finite-dimensional optimization problem. An illustrative example is considered.

Electric transmission lines: Control and numerical discretization

We are interested in the mathematical modeling and control of electrical power lines. The electric transmission lines under consideration are modeled as directed graph. The evolution of voltage and current is described by a linear 2 2 system of hyperbolic balance laws. Open and closed loop control concepts are investigated for control actions at nodal points in the network. For the modeling of electric transmission lines several mathematical approaches can be found in the literature [1, 3]. A typical control problem is to find the optimal power flow when allowing for power supply at selected nodes. Mathematically, this leads to possibly large scale finite-dimensional nonlinear optimization problem [4]. From those, simplified models are derived, which use linear or mixed-integer programming techniques to determine the optimal flow [7]. Based on those models extensions to oscillator networks are used to capture some effects of time-dependent transmission lines. A typical model for the evolution is given by the spatially one-dimensional telegraph equation supplemented by suitable source terms. For instance, in [9] the power and signal losses in communication systems are determined by the telegraph equation. Depending on the sheeted material sometimes lossless transmission lines may be considered. In this case the telegraph equation reduces to the linear wave equation for the voltage. Mathematically, closed loop control questions for the wave equation have extensively been studied for example in [10]. Closed loop control of a single transmission line including source terms has been established using a Lyapunov function approach. We extend the existing result to more general networks and introduce coupling conditions for the telegraph equation as well. The idea is to derive a closed loop network control relying heavily on the construction of the suitable Lyapunov function for the telegraph equation. Furthermore, we introduce an open loop control concept and numerically compare the arising control actions. This extends recent investigations conducted for other applications [3, 11, 12].

Bayesian D-optimal Accelerated Life Test plans for series systems with competing exponential causes of failure

This paper provides methods of obtaining Bayesian D-optimal Accelerated Life Test (ALT) plans for series systems with independent exponential component lives under the Type-I censoring scheme. Two different Bayesian D-optimality design criteria are considered. For both the criteria, first optimal designs for a given number of experimental points are found by solving a finite-dimensional constrained optimization problem. Next, the global optimality of such an ALT plan is ensured by applying the General Equivalence Theorem. A detailed sensitivity analysis is also carried out to investigate the effect of different planning inputs on the resulting optimal ALT plans. Furthermore, these Bayesian optimal plans are also compared with the corresponding (frequentist) locally D-optimal ALT plans.

Application of a Derivative-Free Method with Projection Skill to Solve an Optimization Problem

AbstractImproving numerical forecasting skill in the atmospheric and oceanic sciences by solving optimization problems is an important issue. One such method is to compute the conditional nonlinear optimal perturbation (CNOP), which has been applied widely in predictability studies. In this study, the Differential Evolution (DE) algorithm, which is a derivative-free algorithm and has been applied to obtain CNOPs for exploring the uncertainty of terrestrial ecosystem processes, was employed to obtain the CNOPs for finite-dimensional optimization problems with ball constraint conditions using Burgers’ equation. The aim was first to test if the CNOP calculated by the DE algorithm is similar to that computed by traditional optimization algorithms, such as the Spectral Projected Gradient (SPG2) algorithm. The second motive was to supply a possible route through which the CNOP approach can be applied in predictability studies in the atmospheric and oceanic sciences without obtaining a model adjoint system, or f...

Optimal control with connected initial and terminal conditions

An optimal control problem with linear dynamics is considered on a fixed time interval. The ends of the interval correspond to terminal spaces, and a finite-dimensional optimization problem is formulated on the Cartesian product of these spaces. Two components of the solution of this problem define the initial and terminal conditions for the controlled dynamics. The dynamics in the optimal control problem is treated as an equality constraint. The controls are assumed to be bounded in the norm of L2. A saddle-point method is proposed to solve the problem. The method is based on finding saddle points of the Lagrangian. The weak convergence of the method in controls and its strong convergence in state trajectories, dual trajectories, and terminal variables are proved.

An approach to determining the hydraulic resistance coefficient of a pipeline section under an unsteady flow regime

The study is devoted to determining the value of the hydraulic resistance coefficient of a linear section of a pipeline. The problem is reduced to a problem of finite-dimensional optimization for whose solution we suggest to use the first-order numerical methods. The formulas are obtained for the components of the gradient of the objective functional in the space of parameters to be identified. The results of numerical experiments are presented.

Efficiency through variational-like inequalities with Lipschitz functions

In this work, first we introduce several notions of invexity and pseudoinvexity for a locally Lipschitz function by means of the generalized Jacobian. We study relationships between these concepts, in particular the implications between preinvexity and invexity. Next, we obtain necessary and sufficient optimality conditions for efficient and weak efficient solutions of finite-dimensional (non necessarily Pareto) vector optimization problems with locally Lipschitz objective functions through solutions of vector variational-like inequality problems. These conditions are stated via the generalized Jacobian and under pseudoinvexity hypotheses, and they show that a vector optimization problem can be reformulated as a vector variational-like inequality problem. This work extends and improves several previous papers, where the objective function of the vector optimization problem is assumed to be differentiable, or being locally Lipschitz, the authors consider the componentwise subdifferential based on the Clarke’s generalized gradients of the components of the objective function. Throughout the paper some simple examples are given in order to illustrate the main concepts and results.

Fast Wavelet-Based Model Predictive Control of Differentially Flat Systems

A system is differentially flat if it is Lie–Bäcklund (L-B) equivalent to a free dynamical system that has dimensions equal to that of the input of the original system. Utilizing this equivalence, the problem of nonlinear model predictive control of a flat system can be reduced to a lower dimensional nonlinear programming problem with respect to the flat outputs. In this work, a novel computational method based on Haar wavelets in the time-domain for solving the resulting nonlinear programming problem is developed to obtain an approximation of the optimal flat output trajectory. The Haar wavelet integral operational matrix is utilized to transform the nonlinear programming problem to a finite dimensional nonlinear optimization problem. The proposed approach makes use of flatness as a structural property of nonlinear systems and the convenient mathematical properties of Haar wavelets to develop an efficient computational algorithm for nonlinear model predictive control of differentially flat systems. Further improvement on computational efficiency is achieved by providing solutions with multiple resolutions (e.g., obtaining high resolution solutions only for the near future, but allowing coarse approximation for the later stage in the prediction horizon).

Non-degenerate necessary optimality conditions for the optimal control problem with equality-type state constraints

In this work, an optimal control problem with state constraints of equality type is considered. Novelty of the problem formulation is justified. Under various regularity assumptions imposed on the optimal trajectory, a non-degenerate Pontryagin Maximum Principle is proven. As a consequence of the maximum principle, the Euler---Lagrange and Legendre conditions for a variational problem with equality and inequality state constraints are obtained. As an application, the equation of the geodesic curve for a complex domain is derived. In control theory, the Maximum Principle suggests the global maximum condition, also known as the Weierstrass---Pontryagin maximum condition, due to which the optimal control function, at each instant of time, turns out to be a solution to a global finite-dimensional optimization problem.

Safety Analysis of Stochastic Dynamical Systems

This paper presents a method for verifying the safety of a stochastic system. In particular, we show how to compute the largest set of initial conditions such that a given stochastic system is safe with probability p.To compute the set of initial conditions we rely on the moment method that via Haviland's theorem allows an infinite dimensional optimization problem on measures to be formulated as a polynomial optimization problem. Subsequently, the moment sequence is truncated (relaxed) to obtain a finite dimensional polynomial optimization problem. Finally, we provide an illustrative example that shows how the p-safe initial set is computed numerically.

Convex Optimal Uncertainty Quantification

Optimal uncertainty quantification (OUQ) is a framework for numerical extreme-case analysis of stochastic systems with imperfect knowledge of the underlying probability distribution. This paper presents sufficient conditions under which an OUQ problem can be reformulated as a finite-dimensional convex optimization problem, for which efficient numerical solutions can be obtained. The sufficient conditions include that the objective function is piecewise concave and the constraints are piecewise convex. In particular, we show that piecewise concave objective functions may appear in applications where the objective is defined by the optimal value of a parameterized linear program.

A computational method for solving time-delay optimal control problems with free terminal time

This paper considers a class of optimal control problems for general nonlinear time-delay systems with free terminal time. We first show that for this class of problems, the well-known time-scaling transformation for mapping the free time horizon into a fixed time interval yields a new time-delay system in which the time delays are variable. Then, we introduce a control parameterization scheme to approximate the control variables in the new system by piecewise-constant functions. This yields an approximate finite-dimensional optimization problem with three types of decision variables: the control heights, the control switching times, and the terminal time in the original system (which influences the variable time delays in the new system). We develop a gradient-based optimization approach for solving this approximate problem. Simulation results are also provided to demonstrate the effectiveness of the proposed approach.

Optimal uncertainty quantification with model uncertainty and legacy data

We present an optimal uncertainty quantification (OUQ) protocol for systems that are characterized by an existing physics-based model and for which only legacy data is available, i.e., no additional experimental testing of the system is possible. Specifically, the OUQ strategy developed in this work consists of using the legacy data to establish, in a probabilistic sense, the level of error of the model, or modeling error, and to subsequently use the validated model as a basis for the determination of probabilities of outcomes. The quantification of modeling uncertainty specifically establishes, to a specified confidence, the probability that the actual response of the system lies within a certain distance of the model. Once the extent of model uncertainty has been established in this manner, the model can be conveniently used to stand in for the actual or empirical response of the system in order to compute probabilities of outcomes. To this end, we resort to the OUQ reduction theorem of Owhadi et al. (2013) in order to reduce the computation of optimal upper and lower bounds on probabilities of outcomes to a finite-dimensional optimization problem. We illustrate the resulting UQ protocol by means of an application concerned with the response to hypervelocity impact of 6061-T6 Aluminum plates by Nylon 6/6 impactors at impact velocities in the range of 5–7km/s. The ability of the legacy OUQ protocol to process diverse information on the system and its ability to supply rigorous bounds on system performance under realistic—and less than ideal—scenarios demonstrated by the hypervelocity impact application is remarkable.

The Canonical Theory of the Impulse Process Optimality

The paper is devoted to the development of the canonical theory of the Hamilton–Jacobi optimality for nonlinear dynamical systems with controls of the vector measure type and with trajectories of bounded variation. Infinitesimal conditions of the strong and weak monotonicity of continuous Lyapunov-type functions with respect to the impulsive dynamical system are formulated. Necessary and sufficient conditions of the global optimality for the problem of the optimal impulsive control with general end restrictions are represented. The conditions include the sets of weak and strong monotone Lyapunov-type functions and are based on the reduction of the original problem of the optimal impulsive control a finite-dimensional optimization problem on an estimated set of connectable points.

Gain-Scheduledℋ2Controller Synthesis for Continuous-Time Polytopic LPV Systems

This paper is concerned with the problem of gain-scheduledℋ2controller synthesis for continuous-time linear parameter-varying systems. In this problem, the system matrices in the state-space form are polytopic and patameterized and the admissible values of the parameters are assumed to be measurable on-line in a polytope space. By employing a basis-parameter-dependent Lyapunov function and introducing some slack variables to the well-established performance conditions, sufficient conditions for the existence of the desired gain-scheduledℋ2state feedback and dynamic output feedback controllers are established in terms of parameterized linear matrix inequalities. Based on the polytopic characteristic of the dependent parameters and a convexification method, the corresponding controller synthesis problem is then cast into finite-dimensional convex optimization problem which can be efficiently solved by using standard numerical softwares. Numerical examples are given to illustrate the effectiveness and advantage of the proposed methods.