Target problems under state constraints for nonlinear controlled impulsive systems

We study the problem of reaching a target without leaving a prescribed constraint set for a dynamical system described by a controlled differential equation and a controlled instantaneous reset function. We characterize all initial conditions from which the objective can be reached. Then we characterize the value function associated with the optimal reaching time problem.     

Suboptimal target control for hybrid automata using model predictive control

In this paper, a computationally efficient controller is proposed for the target control problem when the system is modelled by hybrid automata. The design is carried out in two stages. First, we compute off-line the shortest switching path which has the minimum discrete cost from an initial set to the given target set. Next, a controller is derived which successfully drives the system from any given initial state in the initial set to the target set while minimizing a cost function. The model predictive control (MPC) technique is used when the current state is not within a guard set, otherwise the mixed-integer predictive control (MIPC) technique is employed. An on-line, semi-explicit control algorithm is derived by combining these two techniques. When the system is subject to additive bounded disturbance, it is shown that the proposed on-line control algorithm holds if tighter constraints on the original nominal state and controller are imposed. Finally, as an application of the proposed control procedure, the high-speed and energy-saving control problem of the CPU processing is considered.     

关于Hamilton-Jacobi方程解的正则性和全局结构的注记——纪念李国平院士吴新谋教授诞辰100周年

该文考虑高维Hamilton-Jacobi方程的柯西问题. 作者证明了从任一初始点出发的特征线永不碰到奇异点集合的充分必要条件是初始函数在该点取到最小值.在此基础上,证明了奇异点集合的道路连通分支和初始函数不取最小值的点集合的道路连通分支之间存在一一对应, 而且解的梯度的间断一旦产生就不会消失. 特别指出, 该文的结果不需要“初始函数的梯度在无穷远趋近于零”这一限制条件, 而文献[12]中重要的命题2.7和主要结果之一的定理3.3是在这一条件下得到的.     

Direct solution to constrained tropical optimization problems with application to project scheduling

We examine a new optimization problem formulated in the tropical mathematics setting as a further extension of certain known problems. The problem is to minimize a nonlinear objective function, which is defined on vectors over an idempotent semifield by using multiplicative conjugate transposition, subject to inequality constraints. As compared to the known problems, the new one has a more general objective function and additional constraints. We provide a complete solution in an explicit form to the problem by using an approach that introduces an auxiliary variable to represent the values of the objective function, and then reduces the initial problem to a parametrized vector inequality. The minimum of the objective function is evaluated by applying the existence conditions for the solution of this inequality. A complete solution to the problem is given by solving the parametrized inequality, provided the parameter is set to the minimum value. As a consequence, we obtain solutions to new special cases of the general problem. To illustrate the application of the results, we solve a real-world problem drawn from time-constrained project scheduling, and offer a representative numerical example.     

Automated knowledge source selection and service composition

We introduce a new combinatorial problem referred to as the component set identification problem, arising in the context of knowledge discovery, information integration, and knowledge source/service composition. The main motivation for studying this problem is the widespread proliferation of digital knowledge sources and services. Considering a granular knowledge domain consisting of a large number of individual bits and pieces of domain knowledge (properties) and a large number of knowledge sources and services that provide mappings between sets of properties, the objective of the component set identification problem is to select a minimum cost combination of knowledge sources that can provide a joint mapping from a given set of initially available properties (initial knowledge) to a set of initially unknown properties (target knowledge). We position the component set identification problem relative to other combinatorial problems and provide a classification scheme for the different variations of the problem. The problem is next modeled on a directed graph and analyzed in terms of its complexity. The directed graph representation is then augmented and transformed into a time-expanded network representation that is subsequently used to develop an exact solution procedure based on integer programming and branch-and-bound. We enhance the solver by developing preprocessing techniques. All findings are supported by computational experiments.     

A numerical scheme for the impulse control formulation for pricing variable annuities with a guaranteed minimum withdrawal benefit (GMWB)

In this paper, we outline an impulse stochastic control formulation for pricing variable annuities with a guaranteed minimum withdrawal benefit (GMWB) assuming the policyholder is allowed to withdraw funds continuously. We develop a numerical scheme for solving the Hamilton–Jacobi–Bellman (HJB) variational inequality corresponding to the impulse control problem. We prove the convergence of our scheme to the viscosity solution of the continuous withdrawal problem, provided a strong comparison result holds. The scheme can be easily generalized to price discrete withdrawal contracts. Numerical experiments are conducted, which show a region where the optimal control appears to be non-unique.     

Sustainable production technologies which take into account environmental constraints

We formulate and study a multiobjective programming approach for production processes which implements suitable constraints on pollutant emissions. We consider two alternative optimization problems: (a) minimum pollution risk; (b) maximum expected return. For each pollutant, we define three different contamination levels: (a) the desirable or the target pollution level, (b) the alarm (warning or critical) level and (c) the maximum admissible (acceptable) level, and introduce penalties proportional to the amounts of pollutants that exceed these levels. The objective function of the minimum pollution risk problem is not smooth since it contains positive parts of some affine functions, resulting in mathematical difficulties, which can be solved by formulating an alternative linear programming model, which makes use of additional variables and has the same solutions as the initial problem. We investigate various particular cases and analyze a numerical example for a textile plant.     

Optimal impulse and regular control strategies for proportional reinsurance problem

We formulate a stochastic control problem on proportional reinsurance that includes impulse and regular control strategies. For the first time we combine impulse control with regular control, and derive the expected total discount pay-out (return function) from present to bankruptcy. By relying on both stochastic calculus and the classical theory of impulse and regular controls, we state a set of sufficient conditions for its solution in terms of optimal return function. Moreover, we also derive its explicit form and corresponding impulse and regular control strategies.     

Lower semi-continuity of the solution set for semilinear differential inclusions

In this paper we provide sufficient conditions for the solution set of a semilinear differential inclusion to be lower semi-continuous. We give an application to the propagation of continuity of the minimum time function associated with the given differential inclusion and the target zero.     

Robust maximum weighted independent-set problems on interval graphs

We study the maximum weighted independent-set problem on interval graphs with uncertainty on the vertex weights. We use the absolute robustness criterion and the min–max regret criterion to evaluate solutions. For a discrete scenario set, we find that the problem is NP-hard for each of the robustness criteria; we also provide pseudo-polynomial time algorithms when there is a constant number of scenarios and show that the problem is strongly NP-hard when the set of scenarios is unbounded. When the scenario set is a Cartesian product, we prove that the problem is equivalent to a maximum weighted independent-set problem on the same interval graph but without uncertainty for the first objective function and that the scenario set can be reduced for the second objective function.     

Optimal Synthesis for the Minimum Time Control Problems of Fed-Batch Bioprocesses for Growth Functions with Two Maxima

We address the problem of finding an optimal feedback control for feeding a fed-batch bioreactor with one species and one substrate from a given initial condition to a given target value in a minimal amount of time. Recently, the optimal synthesis (optimal feeding strategy) has been obtained in systems in which the microorganisms involved are represented by increasing growth functions or growth functions with one maxima, with either Monod or Haldane functions, respectively (widely used in bioprocesses modeling). In the present work, we allow impulsive controls corresponding to instantaneous dilutions, and we assume that the growth function of the microorganism present in the process has exactly two local maxima. This problem has been tackled from a numerical point of view without impulsive controls. In this article, we introduce two singular arc feeding strategies, and we define explicit regions of initial conditions in which the optimal strategy is either the first singular arc strategy or the second strategy.     

A variational approach for restoring images corrupted by noisy blur kernels and additive noise

In this paper, we study a deblurring algorithm for distorted images by random impulse response. We propose and develop a convex optimization model to recover the underlying image and the blurring function simultaneously. The objective function is composed of 3 terms: the data‐fitting term between the observed image and the product of the estimated blurring function and the estimated image, the squared difference between the estimated blurring function and its mean, and the total variation regularization term for the estimated image. We theoretically show that under some mild conditions, the resulting objective function can be convex in which the global minimum value is unique. The numerical results confirm that the peak‐to‐signal‐noise‐ratio and structural similarity of the restored images by the proposed algorithm are the best when the proposed objective function is convex. We also present a proximal alternating minimization scheme to solve the resulting minimization problem. Numerical examples are presented to demonstrate the effectiveness of the proposed model and the efficiency of the numerical scheme.     

Utility efficiency and its approximation

Summary We assume a multi-attribute decision making problem under uncertainty with partial information on the decision maker's preferences, by a vector utility function with two components and imprecision over their scaling constants. We propose an approximation set whose determination may be easier than the one of the utility efficient set and we consider an interactive procedure which uses such approximation to decision aid. We study some nesting and convergence properties based on the interactive reduction of the approximation set. Finally, we illustrate the procedure with a numerical example.     

Nonlinear mapping and distance geometry

Distance Geometry Problem (DGP) and Nonlinear Mapping (NLM) are two well established questions: DGP is about finding a Euclidean realization of an incomplete set of distances in a Euclidean space, whereas Nonlinear Mapping is a weighted Least Square Scaling (LSS) method. We show how all these methods (LSS, NLM, DGP) can be assembled in a common framework, being each identified as an instance of an optimization problem with a choice of a weight matrix. We study the continuity between the solutions (which are point clouds) when the weight matrix varies, and the compactness of the set of solutions (after centering). We finally study a numerical example, showing that solving the optimization problem is far from being simple and that the numerical solution for a given procedure may be trapped in a local minimum.

     

Eigenfrequencies of a tube bundle immersed in a fluid

In this paper we study a simplified version of a mathematical model that describes the eigenfrequencies and eigenmotions of a coupled system consisting of a set of tubes (or a tube bundle) immersed in an incompressible perfect fluid. The fluid is assumed to be contained in a rectangular cavity, and the tubes are assumed to be identical, and periodically distributed in the cavity. The mathematical model that governs this physical problem is an elliptic differential eigenvalue problem consisting of the Laplace equation with a nonlocal boundary condition on the holes, and a homogeneous Neumann boundary condition on the boundary of the cavity. In the simplified model that we study in this paper, the Neumann condition is replaced by a periodic boundary condition. Our goal in studying this simple version is to derive some basic properties of the problem that could serve as a guide to envisage similar properties for the original model. In practical situations, this kind of problem needs to be solved for tube bundles containing a very large number of tubes. Then the numerical analysis of these problems is in practice very expensive. Several approaches to overcome this difficulty have been proposed in the last years using homogenization techniques. Alternatively, we propose in this paper an approach that consists in obtaining an explicit decomposition of the problem into a finite family of subproblems, which can be easily solved numerically. Our study is based on a generalized notion of periodic function, and on a decomposition theorem for periodic functions that we introduce in the paper. Our results rely on the theory of almost periodic functions, and they provide a simple numerical method for obtaining approximations of all the eigenvalues of the problem for any number of tubes in the cavity. We also discuss a numerical example.     

The Maximin-Maxisum Network Location Problem

In this paper, we consider the problem of finding a point on a general network using two objectives, maximizing the minimum weighted distance from the point to the vertices (Maximin) and maximizing the sum of weighted distances between the point and the vertices (Maxisum). This bicriterion model can be used to locate an obnoxious facility on a network. We will identify the model properties, develop a polynomial algorithm for generating the efficient set and provide a numerical example.     

Rescheduling unrelated parallel machines with total flow time and total disruption cost criteria

In this paper, we consider a rescheduling problem where a set of jobs has already been assigned to unrelated parallel machines. When a disruption occurs on one of the machines, the affected jobs are rescheduled, considering the efficiency and the schedule deviation measures. The efficiency measure is the total flow time, and the schedule deviation measure is the total disruption cost caused by the differences between the initial and current schedules. We provide polynomial-time solution methods to the following hierarchical optimization problems: minimizing total disruption cost among the minimum total flow time schedules and minimizing total flow time among the minimum total disruption cost schedules. We propose exponential-time algorithms to generate all efficient solutions and to minimize a specified function of the measures. Our extensive computational tests on large size problem instances have revealed that our optimization algorithm finds the best solution by generating only a small portion of all efficient solutions.     

Optimality Conditions and Synthesis for the Minimum Time Problem

We study the minimum time optimal control problem for a nonlinear system in R ⁿ with a general target. Necessary and sufficient optimality conditions are obtained. In particular, we describe a class of costates that are included in the superdifferential of the minimum time function, even in the case when this function is only lower semicontinuous. Two set-valued maps are constructed to provide time optimal synthesis.     

A Class of Solvable Impulse Control Problems

We consider a class of stochastic impulse control problems where the controlled process evolves according to a linear, regular, and time homogeneous diffusion. We state a set of easily verifiable sufficient conditions under which the problem is explicitly solvable. We also state an algebraic equation from which the optimal impulse boundary can be determined and, given this threshold, we present the value of the optimal policy in terms of the minimal increasing r-excessive mapping for the controlled diffusion. We also consider the comparative static properties of the optimal policy and state a set of typically satisfied conditions under which increased volatility unambiguously increases the value of the optimal policy and expands the continuation region where exercising the irreversible policy is suboptimal. We also illustrate our results explicitly in two models based on geometric Brownian motion.