The interacting-particle algorithm with dynamic heating and cooling

We consider an interacting-particle algorithm which is population-based like genetic algorithms and also has a temperature parameter analogous to simulated annealing. The temperature parameter of the interacting-particle algorithm has to cool down to zero in order to achieve convergence towards global optima. The way this temperature parameter is tuned affects the performance of the search process and we implement a meta-control methodology that adapts the temperature to the observed state of the samplings. The main idea is to solve an optimal control problem where the heating/cooling rate of the temperature parameter is the control variable. The criterion of the optimal control problem consists of user defined performance measures for the probability density function of the particles’ locations including expected objective function value of the particles and the spread of the particles’ locations. Our numerical results indicate that with this control methodology the temperature fluctuates (both heating and cooling) during the progress of the algorithm to meet our performance measures. In addition our numerical comparison of the meta-control methodology with classical cooling schedules demonstrate the benefits in employing the meta-control methodology.     

Optimization of Algorithmic Parameters using a Meta-Control Approach*

Optimization algorithms usually rely on the setting of parameters, such as barrier coefficients. We have developed a generic meta-control procedure to optimize the behavior of given iterative optimization algorithms. In this procedure, an optimal continuous control problem is defined to compute the parameters of an iterative algorithm as control variables to achieve a desired behavior of the algorithm (e.g., convergence time, memory resources, and quality of solution). The procedure is illustrated with an interior point algorithm to control barrier coefficients for constrained nonlinear optimization. Three numerical examples are included to demonstrate the enhanced performance of this method. This work was primarily done when Z. Zabinsky was visiting Clearsight Systems Inc.     

The inverse optimal value problem

This paper considers the following inverse optimization problem: given a linear program, a desired optimal objective value, and a set of feasible cost vectors, determine a cost vector such that the corresponding optimal objective value of the linear program is closest to the desired value. The above problem, referred here as the inverse optimal value problem, is significantly different from standard inverse optimization problems that involve determining a cost vector for a linear program such that a pre-specified solution vector is optimal. In this paper, we show that the inverse optimal value problem is NP-hard in general. We identify conditions under which the problem reduces to a concave maximization or a concave minimization problem. We provide sufficient conditions under which the associated concave minimization problem and, correspondingly, the inverse optimal value problem is polynomially solvable. For the case when the set of feasible cost vectors is polyhedral, we describe an algorithm for the inverse optimal value problem based on solving linear and bilinear programming problems. Some preliminary computational experience is reported.Mathematics Subject Classification (1999):49N45, 90C05, 90C25, 90C26, 90C31, 90C60Acknowledgement This research has been supported in part by the National Science Foundation under CAREER Award DMII-0133943. The authors thank two anonymous reviewers for valuable comments.     

Constructive Solution of Inverse Parametric Linear/Quadratic Programming Problems

Parametric convex programming has received a lot of attention, since it has many applications in chemical engineering, control engineering, signal processing, etc. Further, inverse optimality plays an important role in many contexts, e.g., image processing, motion planning. This paper introduces a constructive solution of the inverse optimality problem for the class of continuous piecewise affine functions. The main idea is based on the convex lifting concept. Accordingly, an algorithm to construct convex liftings of a given convexly liftable partition will be put forward. Following this idea, an important result will be presented in this article: Any continuous piecewise affine function defined over a polytopic partition is the solution of a parametric linear/quadratic programming problem. Regarding linear optimal control, it will be shown that any continuous piecewise affine control law can be obtained via a linear optimal control problem with the control horizon at most equal to 2 prediction steps.     

Experimental design for distributed parameter vector systems

We formulate an optimal design problem for the selection of best states to observe and optimal sampling times and locations for parameter estimation or inverse problems involving complex nonlinear partial differential systems. An iterative algorithm for implementation of the resulting methodology is proposed.     

Adaptive Inverse Control Using a Gradient-Projection Optimization Technique

As an application of an optimization technique, a gradient-projection method is employed to derive an adaptive algorithm for updating the parameters of an inverse which is designed to cancel the effects of actuator uncertainties in a control system. The actuator uncertainty is parametrized by a set of unknown parameters which belong to a parameter region. A desirable inverse is implemented with adaptive estimates of the actuator parameters. Minimizing an estimation error, a gradient algorithm is used to update such parameter estimates. To ensure that the parameter estimates also belong to the parameter region, the adaptive update law is designed with parameter projection. With such an adaptive inverse, desired control system performance can be achieved despite the presence of the actuator uncertainties.     

Parameter identification in financial market models with a feasible point SQP algorithm

The quickly moving market data in the finance industry requires a frequent parameter identification of the corresponding financial market models. In this paper we apply a special sequential quadratic programming algorithm to the calibration of typical equity market models. As it turns out, the projection of the iterates onto the feasible set can be efficiently computed by solving a semidefinite programming problem. Combining this approach with a Gauss-Newton framework leads to an efficient algorithm which allows to calibrate e.g. Heston’s stochastic volatility model in less than a half second on a usual 3 GHz desktop PC. Furthermore we present an appropriate regularization technique that stabilizes and significantly speeds up computations if the model parameters are chosen to be time-dependent.     

An optimal control problem in estimating the cooling condition for a cemented hip replacement system

An optimal control problem utilizing the Levenberg–Marquardt method (LMM) is examined in this study to determine the unknown optimal control heat flux function for a cemented hip replacement system based on the desired temperature distributions at the cement–bone interface to prevent the death of bone tissues. The validation of this optimal control problem is verified by using the numerical experiments. Results show that an optimal control function can be obtained using the present algorithm for the test cases considered in this work to reduce the temperature variation and to save the bone tissues at the cement–bone interface.     

Robust and efficient fitting of the generalized Pareto distribution with actuarial applications in view

Due to advances in extreme value theory, the generalized Pareto distribution (GPD) emerged as a natural family for modeling exceedances over a high threshold. Its importance in applications (e.g., insurance, finance, economics, engineering and numerous other fields) can hardly be overstated and is widely documented. However, despite the sound theoretical basis and wide applicability, fitting of this distribution in practice is not a trivial exercise. Traditional methods such as maximum likelihood and method-of-moments are undefined in some regions of the parameter space. Alternative approaches exist but they lack either robustness (e.g., probability-weighted moments) or efficiency (e.g., method-of-medians), or present significant numerical problems (e.g., minimum-divergence procedures). In this article, we propose a computationally tractable method for fitting the GPD, which is applicable for all parameter values and offers competitive trade-offs between robustness and efficiency. The method is based on ‘trimmed moments’. Large-sample properties of the new estimators are provided, and their small-sample behavior under several scenarios of data contamination is investigated through simulations. We also study the effect of our methodology on actuarial applications. In particular, using the new approach, we fit the GPD to the Danish insurance data and apply the fitted model to a few risk measurement and ratemaking exercises.     

The dual weighted residuals approach to optimal control of ordinary differential equations

The methodology of dual weighted residuals is applied to an optimal control problem for ordinary differential equations. The differential equations are discretized by finite element methods. An a posteriori error estimate is derived and an adaptive algorithm is formulated. The algorithm is implemented in Matlab and tested on a simple model problem from vehicle dynamics.     

抑制剂作用下肿瘤生长模型的参数识别

该文考虑抑制剂作用下肿瘤生长的模型. 假设肿瘤是球对称的, 其表面为运动边界, 用函数r=R(t)表示. 既然多细胞肿瘤扁球体(MTS)通常作为肿瘤生长的体外模型, 在实验室能够被观察和控制, 因此研究如下反问题: 根据观察到的MTS动态增长(即给定R(t)), 来确定抑制剂的参数. 运用极大值原理, 作者证明了该抛物反问题解的唯一性. 进一步, 用最优控制框架来重构模型中的抑制剂参数, 证明了最优控制问题解的存在性, 并推导了最优控制满足的最优性必要条件.     

Nonlinear analysis of elastic thin-walled shell structures

Theoretical principles, methodology and algorithms presented herein are to analyze and design the elastic thin-walled engineering structures and components, with emphasis on the important nonlinear behavior. The methodology of the consequent analysis of single-parametric nonlinear problems is applied to structural syntheses. The numerical algorithm for this analysis is based on the parameter continuation methods and the “control parameter subspace changing”. The effectiveness of the proposed approach is illustrated through several examples in thin-walled structures.     

A relaxed cutting plane method for semi-infinite semi-definite programming

In this paper, we develop two discretization algorithms with a cutting plane scheme for solving combined semi-infinite and semi-definite programming problems, i.e., a general algorithm when the parameter set is a compact set and a typical algorithm when the parameter set is a box set in the m-dimensional space. We prove that the accumulation point of the sequence points generated by the two algorithms is an optimal solution of the combined semi-infinite and semi-definite programming problem under suitable assumption conditions. Two examples are given to illustrate the effectiveness of the typical algorithm.     

A solution of the discrete minimum-time control problem

An iterative procedure for the synthesis of discrete minimum-amplitude and minimum-time controls is presented. The algorithm is based on some new relations obtained by extending well-known results on the minimum-energy control problem as given in Refs. 1–3. This approach yields a set of implicit algebraic equations from which the desired optimal control sequence is determined by the iteration procedure referred to above. The algorithm has the advantage that convergence to the optimal solution can be guaranteed. Simplicity of the recursion formulas and insensitivity to numerical errors make the procedure well suited for on-line or off-line computations.This work was done at the Institut für Regelungstechnik, Technische Universität Berlin, West Berlin, Germany. The author is indebted to Professor G. Schneider for many stimulating discussions and criticisms during the course of this research.     

Global multi-parametric optimal value bounds and solution estimates for separable parametric programs

In this tutorial, a strategy is described for calculating parametric piecewise-linear optimal value bounds for nonconvex separable programs containing several parameters restricted to a specified convex set. The methodology is based on first fixing the value of the parameters, then constructing sequences of underestimating and overestimating convex programs whose optimal values respectively increase or decrease to the global optimal value of the original problem. Existing procedures are used for calculating parametric lower bounds on the optimal value of each underestimating problem and parametric upper bounds on the optimal value of each overestimating problem in the sequence, over the given set of parameters. Appropriate updating of the bounds leads to a nondecreasing sequence of lower bounds and a nonincreasing sequence of upper bounds, on the optimal value of the original problem, continuing until the bounds satisfy a specified tolerance at the value of the parameter that was fixed at the outset. If the bounds are also sufficiently tight over the entire set of parameters, according to criteria specified by the user, then the calculation is complete. Otherwise, another parameter value is selected and the procedure is repeated, until the specified criteria are satisfied over the entire set of parameters. A parametric piecewise-linear solution vector approximation is also obtained. Results are expected in the theory, computations, and practical applications. The general idea of developing results for general problems that are limits of results that hold for a sequence of well-behaved (e.g., convex) problems should be quite fruitful.     

Optimal control of bioprocess systems using hybrid numerical optimization algorithms

In the paper, we consider the bioprocess system optimal control problem. Generally speaking, it is very difficult to solve this problem analytically. To obtain the numerical solution, the problem is transformed into a parameter optimization problem with some variable bounds, which can be efficiently solved using any conventional optimization algorithms, e.g. the improved Broyden–Fletcher–Goldfarb–Shanno algorithm. However, in spite of the improved Broyden–Fletcher–Goldfarb–Shanno algorithm is very efficient for local search, the solution obtained is usually a local extremum for non-convex optimal control problems. In order to escape from the local extremum, we develop a novel stochastic search method. By performing a large amount of numerical experiments, we find that the novel stochastic search method is excellent in exploration, while bad in exploitation. In order to improve the exploitation, we propose a hybrid numerical optimization algorithm to solve the problem based on the novel stochastic search method and the improved Broyden–Fletcher–Goldfarb–Shanno algorithm. Convergence results indicate that any global optimal solution of the approximate problem is also a global optimal solution of the original problem. Finally, two bioprocess system optimal control problems illustrate that the hybrid numerical optimization algorithm proposed by us is low time-consuming and obtains a better cost function value than the existing approaches.     

Optimization of the heating of a heat-sensitive layer under constraints on the stresses in the plastic region of deformation of the material

We study the problem of optimal control for rapidity of the heating of a heat-sensitive layer under constraints on the control (the temperature of the heating medium or the heat flux) and maximal values of the stress intensity in the plastic region of deformation of the material. We propose an algorithm for solving the problem that presumes it has been reduced to the inverse problem of thermoplasticity. For the case of one-sided heating we give a numerical analysis of the direct and inverse problems of thermoplasticity. Translated fromMatematichni Metody i Fiziko-Mekhanichni Polya, Vol. 38, 1995.     

Flight control system synthesis using eigenstructure assignment

The central topic of this paper is the establishment of an efficient practical synthesis procedure for modern flight control systems. Unlike the classical design methodology (Bode plots, Nichols plot, etc.) and optimal control techniques, the present approach provides the designer a direct approach for the synthesis of desired control laws. Although the setting is the now familiar state space, the actual design is performed relative to classical specifications (i.e., modes and mode distribution) by placing closed-loop eigenvalues and eigenvectors at some desired locations (regions) within the state space. The new method can handle output feedback configurations, subject to controller structural constraints. Complete theoretical background and a realistic numerical example are provided.     

Multiobjective Optimization of the Flow Around a Cylinder Using Model Order Reduction

In this article an efficient numerical method to solve multiobjective optimization problems for fluid flow governed by the Navier Stokes equations is presented. In order to decrease the computational effort, a reduced order model is introduced using Proper Orthogonal Decomposition and a corresponding Galerkin Projection. A global, derivative free multiobjective optimization algorithm is applied to compute the Pareto set (i.e. the set of optimal compromises) for the concurrent objectives minimization of flow field fluctuations and control cost. The method is illustrated for a 2D flow around a cylinder at Re = 100. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optimal control of time discrete two-phase flow governed by a diffuse interface model

We present results on optimal control of two-phase flows. The fluid is modeled by a thermodynamically consistent diffuse interface model and allows to treat fluids of different densities and viscosities. In earlier work we proposed an energy stable time discretization for this model that we now employ to derive existence of optimal controls for a time discrete optimal control problem. The control aim is to obtain a desired distribution of the two phases in the system. For this we investigate three control actions. We use tangential Dirichlet boundary control and distributed control. We further consider the inverse problem of finding an initial distribution such that the evolution over a given time horizon starting from this value is close to a desired distribution. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)