Efficient Sequential Quadratic Programming Implementations for Equality-Constrained Discrete-Time Optimal Control

Efficient sequential quadratic programming (SQP) implementations are presented for equality-constrained, discrete-time, optimal control problems. The algorithm developed calculates the search direction for the equality-based variant of SQP and is applicable to problems with either fixed or free final time. Problem solutions are obtained by solving iteratively a series of constrained quadratic programs. The number of mathematical operations required for each iteration is proportional to the number of discrete times N. This is contrasted by conventional methods in which this number is proportional to N ³. The algorithm results in quadratic convergence of the iterates under the same conditions as those for SQP and simplifies to an existing dynamic programming approach when there are no constraints and the final time is fixed. A simple test problem and two application problems are presented. The application examples include a satellite dynamics problem and a set of brachistochrone problems involving viscous friction.     

一类经典”秘书问题”的推广

”秘书问题”在最优停时理论的发展中曾起过重要作用 ,实际中的一类问题与”秘书问题”有类似之处 ,但比”秘书问题”更复杂 .本文将经典”秘书问题”进行推广 ,建立了一类比经典”秘书问题”更有实际意义的模型 ,并给出了该类模型的解 .     

Optimal purchase and advertising for a product with immediate sale start

Summary We consider the problem of maximizing the discounted net profit of a firm which purchases a quantity of some product at a given time and afterwards advertises and sells the product progressively. We distinguish among the three possibilities of assuming the final time to be either fixed, or bounded, or free. In all cases, after stating the problem in the optimal control theory framework, we prove the existence of an optimal solution and characterize it using the Maximum Principle necessary conditions. Furthermore, we prove that the convexity of the purchase cost function is a sufficient condition for the uniqueness of the optimal solution. Partially supported by MURST.     

Optimal vaccine scheduling in cancer immunotherapy

Cancer immunotherapy aims at stimulating the immune system to react against cancer stealth capabilities. It consists of repeatedly injecting small doses of a tumor-associated molecule one wants the immune system to recognize, until a consistent immune response directed against the tumor cells is observed.

We have applied the theory of optimal control to the problem of finding the optimal schedule of injections of an immunotherapeutic agent against cancer. The method employed works for a general ODE system and can be applied to find the optimal protocol in a variety of clinical problems where the kinetics of the drug or treatment and its influence on the normal physiologic functions have been described by a mathematical model.

We show that the choice of the cost function has dramatic effects on the kind of solution the optimization algorithm is able to find. This provides evidence that a careful ODE model and optimization schema must be designed by mathematicians and clinicians using their proper different perspectives.     

Normality of the maximum principle for nonconvex constrained Bolza problems

We consider a Bolza optimal control problem with state constraints. It is well known that under some technical assumptions every strong local minimizer of this problem satisfies first order necessary optimality conditions in the form of a constrained maximum principle. In general, the maximum principle may be abnormal or even degenerate and so does not provide a sufficient information about optimal controls. In the recent literature some sufficient conditions were proposed to guarantee that at least one maximum principle is nondegenerate, cf. [A.V. Arutyanov, S.M. Aseev, Investigation of the degeneracy phenomenon of the maximum principle for optimal control problems with state constraints, SIAM J. Control Optim. 35 (1997) 930–952; F. Rampazzo, R.B. Vinter, A theorem on existence of neighbouring trajectories satisfying a state constraint, with applications to optimal control, IMA 16 (4) (1999) 335–351; F. Rampazzo, R.B. Vinter, Degenerate optimal control problems with state constraints, SIAM J. Control Optim. 39 (4) (2000) 989–1007]. Our aim is to show that actually conditions of a similar nature guarantee normality of every nondegenerate maximum principle. In particular we allow the initial condition to be fixed and the state constraints to be nonsmooth. To prove normality we use J. Yorke type linearization of control systems and show the existence of a solution to a linearized control system satisfying new state constraints defined, in turn, by linearization of the original set of constraints along an extremal trajectory.     

Optimal design of a convergent-barrel cold spray nozzle by numerical method

A convergent-barrel (CB) cold spray nozzle was designed through numerical simulation. It was found that the main factors influencing significantly particle velocity and temperature include the length and diameter of the barrel section, the nature of the accelerating gas and its pressure and temperature, and the particle size. Particles can achieve a relatively low velocity but a high temperature under the same gas pressure using a CB nozzle compared to a convergent-divergent (CD) nozzle. The experiment results with Cu powder using the designed CB nozzle confirmed that particle deposition can be realized under a lower gas pressure with a CB nozzle.     

Stratifiable Families of Extremals and Sufficient Conditions for Optimality in Optimal Control Problems

It is shown that, if a parametrized fämily of extremals F can be stratified in a way compatible with the flow map generated by F, then those trajectories of the family which realize the minimal values of the cost in F are indeed optimal in comparison with all trajectories which lie in the region R covered by the trajectories of F. It is not assumed that F is a field covering the state space injectively. As illustration, an optimal synthesis is constructed for a system where the flow of extremals exhibits a simple cusp singularity.     

Maximin share and minimax envy in fair-division problems

For fair-division or cake-cutting problems with value functions which are normalized positive measures (i.e., the values are probability measures) maximin-share and minimax-envy inequalities are derived for both continuous and discrete measures. The tools used include classical and recent basic convexity results, as well as ad hoc constructions. Examples are given to show that the envy-minimizing criterion is not Pareto optimal, even if the values are mutually absolutely continuous. In the discrete measure case, sufficient conditions are obtained to guarantee the existence of envy-free partitions.     

Optimal Control of Magnetohydrodynamic Equations with State Constraint

This work is concerned with the maximum principle for optimal control problem governed by magnetohydrodynamic equations, which describe the motion of a viscous incompressible conducting fluid in a magnetic field and consist of a subtle coupling of the Navier-Stokes equation of viscous incompressible fluid flow and the Maxwell equation of electromagnetic field. An integral type state constraint is considered.     

Memory and a priori best strategy in complex adaptive systems

We employ an agent‐based model to show that memory and the absence of an a priori best strategy are sufficient for self‐segregation and clustering to emerge in a complex adaptive system with discrete agents that do not compete over a limited resource nor contend in a winner‐take‐all scenario. An agent starts from a corner of a two‐dimensional lattice and aims to reach a randomly selected site in the opposite side within the shortest possible time. The agent is isolated during the course of its journey and does not interact with other agents. Time‐bound obstacles appear at random lattice locations and the agent must decide whether to challenge or evade any obstacle blocking its path. The agent is capable of adapting a strategy in dealing with an obstacle. We analyze the dependence of strategy‐retention time with strategy for both memory‐based and memory‐less agents. We derive the equality spectrum to establish the environmental conditions that favor the existence of an a priori best strategy. We found that memory‐less agents do not polarize into two opposite strategy‐retention time distributions nor cluster toward a center distribution. © 2004 Wiley Periodicals, Inc. Complexity 9: 41–46, 2004