$$L^1$$ penalization of volumetric dose objectives in optimal control of PDEs

This work is concerned with a class of PDE-constrained optimization problems that are motivated by an application in radiotherapy treatment planning. Here the primary design objective is to minimize the volume where a functional of the state violates a prescribed level, but prescribing these levels in the form of pointwise state constraints leads to infeasible problems. We therefore propose an alternative approach based on \(L^1\) penalization of the violation that is also applicable when state constraints are infeasible. We establish well-posedness of the corresponding optimal control problem, derive first-order optimality conditions, discuss convergence of minimizers as the penalty parameter tends to infinity, and present a semismooth Newton method for their efficient numerical solution. The performance of this method for a model problem is illustrated and contrasted with an alternative approach based on (regularized) state constraints.     

Variable objective search

This paper introduces the variable objective search framework for combinatorial optimization. The method utilizes different objective functions used in alternative mathematical programming formulations of the same combinatorial optimization problem in an attempt to improve the solutions obtained using each of these formulations individually. The proposed technique is illustrated using alternative quadratic unconstrained binary formulations of the classical maximum independent set problem in graphs.     

Problem of optimal endogenous growth with exhaustible resources and possibility of a technological jump

In 2013, S. Aseev, K. Besov, and S. Kaniovski (“The problem of optimal endogenous growth with exhaustible resources revisited,” Dyn. Model. Econometr. Econ. Finance 14, 3–30) considered the problem of optimal dynamic allocation of economic resources in an endogenous growth model in which both production and research sectors require an exhaustible resource as an input. The problem is formulated as an infinite-horizon optimal control problem with an integral constraint imposed on the control. A full mathematical study of the problem was carried out, and it was shown that the optimal growth is not sustainable under the most natural assumptions about the parameters of the model. In the present paper we extend the model by introducing an additional possibility of “random” transition (jump) to a qualitatively new technological trajectory (to an essentially unlimited backstop resource). As an objective functional to be maximized, we consider the expected value of the sum of the objective functional in the original problem on the time interval before the jump and an evaluation of the state of the model at the moment of the jump. The resulting problem also reduces to an infinite-horizon optimal control problem, and we prove an existence theorem for it and write down an appropriate version of the Pontryagin maximum principle. Then we characterize the optimal transitional dynamics and compare the results with those for the original problem (without a jump).     

Minimizing the variance of a mathematical expectation estimate for a diffusion process functional based on a parametric transformation of a parabolic boundary value problem

This paper deals with finding ways of reducing the variance of a mathematical expectation estimate for the functional of a diffusion process moving in a domain with an absorbing boundary. The estimate of mathematical expectation of the functional is obtained based on a numerical solution of stochastic differential equations (SDEs) by using the Euler method. A formula of the limiting variance is derived with decreasing integration step in the Euler method. A method of reducing the variance value of the estimate based on transformation of the parabolic boundary value problem corresponding to the diffusion process is proposed. Some numerical results are presented.     

A class of nondifferentiable control problems

Optimality conditions and duality results are obtained for a class of control problems having a nondifferentiable term in the integrand of the objective functional. These results generalize many well-known results in optimal control theory involving differentiable functions, and also provide a relationship with certain nondifferentiable mathematical programming problems. Some extensions concerning the unified treatment of optimal control theory and continuous programming are also mentioned. Finally, a control problem containing an arbitrary norm, along with its appropriate norm, is given.     

The impact of sampling methods on bias and variance in stochastic linear programs

Stochastic linear programs can be solved approximately by drawing a subset of all possible random scenarios and solving the problem based on this subset, an approach known as sample average approximation (SAA). The value of the objective function at the optimal solution obtained via SAA provides an estimate of the true optimal objective function value. This estimator is known to be optimistically biased; the expected optimal objective function value for the sampled problem is lower (for minimization problems) than the optimal objective function value for the true problem. We investigate how two alternative sampling methods, antithetic variates (AV) and Latin Hypercube (LH) sampling, affect both the bias and variance, and thus the mean squared error (MSE), of this estimator. For a simple example, we analytically express the reductions in bias and variance obtained by these two alternative sampling methods. For eight test problems from the literature, we computationally investigate the impact of these sampling methods on bias and variance. We find that both sampling methods are effective at reducing mean squared error, with Latin Hypercube sampling outperforming antithetic variates. For our analytic example and the eight test problems we derive or estimate the condition number as defined in Shapiro et al. (Math. Program. 94:1–19, 2002). We find that for ill-conditioned problems, bias plays a larger role in MSE, and AV and LH sampling methods are more likely to reduce bias.     

Conjoint measurement in multiple objective decision making: A new approach

A new methodology for estimating the objective function in a multiple objective mathematical programming model is presented. A decision maker is required to provide pairwise preferences, or rank orders, of a set of solutions to the multiple objective problem. Conjoint measurement is then applied to this preference information to estimate parameters of an assumed utility function. Simulation tests support the method as being mathematically tractable, while a laboratory study with human decision makers provides encouraging results for the practicality of the technique.     

Stabilization for the Kirchhoff type equation from an axially moving heterogeneous string modeling with boundary feedback control

The first objective of this paper is to make the mathematical model for vibration suppression of an axially moving heterogeneous string. In order to describe the geometrical nonlinearity due to finite transverse deformation, the exact expression of the strain is used. The mathematical modeling is derived first by using Hamilton’s principle and variational lemma and the derived nonlinear PDE system is the Kirchhoff type equation with boundary feedback control. Next, we show the existence and uniqueness of strong solutions of the PDE system via techniques of functional analysis, mainly a theorem of compactness for the analysis of the approximation of the Faedo–Galerkin method and estimate a decay rate for the energy. The theoretical results are assured by numerical results of the solution’s shape and asymptotic behavior for the system.     

“Invariance-Inducing” Control of Nonlinear Discrete-Time Dynamical Systems

The present study aims at the derivation of model-based control laws that attain the invariance objective for nonlinear skew-product discrete-time dynamical systems. The problem under consideration naturally arises in a variety of control problems pertaining to physical/chemical systems, and in the present study, it is conveniently formulated and addressed in the context of functional equations theory. In particular, the mathematical formulation of the problem of interest is realized via a system of nonlinear functional equations (NFEs), and a rather general set of necessary and sufficient conditions for solvability is derived. The solution to the above system of NFEs can be proven to be a unique locally analytic one, and this enables the development of a series solution method that is easily programmable with the aid of a symbolic software package such as MAPLE. It is also shown that, on the basis of the solution to the above system of NFEs, a locally analytic manifold and a nonlinear control law can be explicitly derived that renders the manifold invariant for the class of skew-product systems considered. Furthermore, the restriction of the system dynamics on the aforementioned invariant manifold represents exactly the target controlled system dynamics. Finally, the proposed method is applied to the HF molecular system classically modeled as a rotationless Morse oscillator in the presence of an external laser-field, where the primary objective is molecular dissociation.     

Simultaneous Posterior Probability Statements From Monte Carlo Output

This article considers the problem of making simultaneous probability statements in multivariate inferential problems based on samples from a posterior distribution. The calculation of simultaneous credible bands is reviewed and—as an alternative—contour probabilities are proposed. These are defined as 1 minus the content of the highest posterior density region which just covers a certain point of interest. We discuss a Monte Carlo method to estimate contour probabilities and distinguish whether or not the functional form of the posterior density is available. In the latter case, an approach based on Rao-Blackwellization is proposed. We highlight that this new estimate has an important invariance property. We illustrate the performance of the different methods in three applications.     

Conditions for the Absence of Local Extrema in Problems of Quantum Coherent Control

We consider a terminal control problem for quantum systems which is formulated as the problem of maximizing the objective functional at some fixed finite time. Within the framework of this problem, we discuss known results on the local maxima of the objective functional that are not global. This question is important for quantum control, since such local maxima could make it difficult to find the global maximum by local search in numerical optimization or under laboratory conditions.     

Optimal control of plant virus propagation

Plants are a food source for man and many species. But, plants are subject to diseases, many of which are caused by viruses. Usually, virus propagation is done by a vector. Insect vectors typically have a seasonal behavior, and processes have delays. To combat the vectors, chemical insecticides are commonly used as a control. Unfortunately, these chemicals not only are expensive but also have toxic effects on humans, animals, and the environment. An alternative is to introduce a predator species to prey on the insects and limit the spread of the virus. A combination of insecticide and predators can be used to control the vector population. The question is whether there is an optimal combination. We introduce a mathematical model of ordinary differential equations describing the interaction between plants, vectors, and predators. To determine the optimal amount of predators to introduce and insecticide to use, an objective function giving the total cost to the farmer of the disease is given. We find the controls that minimize the objective function subject to the population variables satisfying the differential equation model and initial conditions together with constraints. There are two main different approaches that can be used to solve the optimal control problem: indirect and direct methods. We use direct methods to solve the problem with and without seasonality and delays. From the practical side, the model can be used to help farmers determine the right balance of insecticide and predators to minimize the total cost.     

Optimal control of a reflection boundary coefficient in an acoustic wave equation

We seek to optimally control a reflection boundary coefficient for an acoustic wave equation. The goal-quantified by an objective functional- is to drive the solution close to a target by adjusting this coefficient, which acts as a control. The problem is solved by finding the optimal control, which minimizes the objective functional. Then the optimal control is used as a an approximation for an inverse “ identification” problem.     

Optimal premium policy of an insurance firm: Full and partial information

Herein, we study the optimization problem faced by an insurance firm who can control its cash-balance dynamics by adjusting the underlying premium rate. The firm’s objective is to minimize the total deviation of its cash-balance process to some pre-set target levels by selecting an appropriate premium policy. Our problem is totally new and has three distinguishable features: (1) both full and partial information cases are investigated here; (2) the state is subject to terminal constraint; (3) a forward-backward stochastic differential equation formulation is given which is more systematic and mathematically advanced. This formulation also enables us to continue further research in a generalized stochastic recursive control framework (see Duffie and Epstein (1992), El Karoui et al. (2001), etc.). The optimal premium policy with the associated optimal objective functional are completely and explicitly derived. In addition, a backward separation technique adaptive to forward-backward stochastic systems with the state constraint is presented as an efficient and convenient alternative to the traditional Wonham’s (1968) separation principle in our partial information setup. Some concluding remarks are also given here.     

Spectral Optimization Methods for the Time Fractional Diffusion Inverse Problem

An inverse problem of reconstructing the initial condition for a time fractional diffusion equation is investigated. On the basis of the optimal control framework, the uniqueness and first order necessary optimality condition of the minimizer for the objective functional are established, and a time-space spectral method is proposed to numerically solve the resulting minimization problem. The contribution of the paper is threefold: 1) a priori error estimate for the spectral approximation is derived; 2) a conjugate gradient optimization algorithm is designed to efficiently solve the inverse problem; 3) some numerical experiments are carried out to show that the proposed method is capable to find out the optimal initial condition, and that the convergence rate of the method is exponential if the optimal initial condition is smooth.     

Optimal Control of Cancer Chemotherapy with Delays and State Constraints

Journal of Optimization Theory and Applications - A mathematical model of cancer chemotherapy is considered as an optimal control problem with the objective of either minimizing a weighted sum of...     

Some integer programs arising in the design of main frame computers

In this paper we describe and discuss a problem that arises in the (global) design of a main frame computer. The task is to assign certain functional units to a given number of so called multi chip modules or printed circuit boards taking into account many technical constraints and minimizing a complex objective function. We describe the real world problem. A thorough mathematical modelling of all aspects of this problem results in a rather complicated integer program that seems to be hopelessly difficult — at least for the present state of integer programming technology. We introduce several relaxations of the general model, which are alsoNP-hard, but seem to be more easily accessible. The mathematical relations between the relaxations and the exact formulation of the problem are discussed as well.On leave from University of São Paulo, Brazil.     

On necessary optimality conditions for infinite-horizon economic growth problems with locally unbounded instantaneous utility function

We consider a class of infinite-horizon optimal control problems that arise in studying models of optimal dynamic allocation of economic resources. In a typical problem of that kind the initial state is fixed, no constraints are imposed on the behavior of the admissible trajectories at infinity, and the objective functional is given by a discounted improper integral. Earlier, for such problems, S.M. Aseev and A.V. Kryazhimskiy in 2004–2007 and jointly with the author in 2012 developed a method of finite-horizon approximations and obtained variants of the Pontryagin maximum principle that guarantee normality of the problem and contain an explicit formula for the adjoint variable. In the present paper those results are extended to a more general situation where the instantaneous utility function need not be locally bounded from below. As an important illustrative example, we carry out a rigorous mathematical investigation of the transitional dynamics in the neoclassical model of optimal economic growth.     

Project selection,scheduling and resource allocation with time dependent returns

In this paper we formulate and analyze the joint problem of project selection and task scheduling. We study the situation where a manager has many alternative projects to pursue such as developing new product platforms or technologies, incremental product upgrades, or continuing education of human resources. Project return is assumed to be a known function of project completion time. Resources are limited and renewable. The objective is to maximize present worth of profit. A general mathematical formulation that can address several versions of the problem is presented. An implicit enumeration procedure is then developed and tested to provide good solutions based on project ordering and a prioritization rule for resource allocation. The algorithm uses an imbedded module for solving the resource-constrained project scheduling problem at each stage. The importance of integrating the impact of resource constraints into the selection of projects is demonstrated.     

Sensitivity analysis and model optimization for reaction-diffusion systems

This paper presents a computational framework for the optimization and sensitivity analysis of a process whose state depends upon several parameter functions. Assuming that the process is described by a system of quasilinear, parabolic, partial differential equations, we show how determining the problem parameters so as to improve an associated objective functional is directly related to knowing the state function sensitivities. An expression for the gradient of the objective functional in terms of the solutions of an adjoint system enables one to bypass the calculation of state function sensitivities. These concepts are illustrated for a simple model of cooperative processes in chemical kinetics. Since sensitivity analysis and model optimization are important tools for investigating parameter dependence and validating mathematical models, research developments in such diverse fields as optimal design theory, chemical kinetics, and parameter identification are important motivations for this paper.This author would like to gratefully acknowledge Dr. M. Delle Donne, EGG, for several helpful discussions.This author was partially supported by NSF Grant No. CMS-80-05677.