C. Yalçın Kaya

Transition probability bounds for the stochastic stability robustness of continuous- and discrete-time Markovian jump linear systems

This paper considers the robustness of stochastic stability of Markovian jump linear systems in continuous- and discrete-time with respect to their transition rates and probabilities, respectively. The continuous-time (discrete-time) system is described via a continuous-valued state vector and a discrete-valued mode which varies according to a Markov process (chain). By using stochastic Lyapunov function approach and Kronecker product transformation techniques, sufficient conditions are obtained for the robust stochastic stability of the underlying systems, which are in terms of upper bounds on the perturbed transition rates and probabilities. Analytical expressions are derived for scalar systems, which are straightforward to use. Numerical examples are presented to show the potential of the proposed techniques.

Inexact Restoration for Runge-Kutta Discretization of Optimal Control Problems

A numerical method is presented for Runge-Kutta discretization of unconstrained optimal control problems. First, general Runge-Kutta discretization is carried out to obtain a finite-dimensional approximation of the continous-time optimal control problem. Then a recent optimization technique, the inexact restoration (IR) method, due to Martinez and coworkers [E. G. Birgin and J. M. Martinez, J. Optim. Theory Appl., 127 (2005), pp. 229-247; J. M. Martinez and E. A. Pilotta, J. Optim. Theory Appl., 104 (2000), pp. 135-163; J. M. Martinez, J. Optim. Theory Appl., 111 (2001), pp. 39-58], is applied to the discretized problem to find an approximate solution. It is proved that, for optimal control problems, a key sufficiency condition for convergence of the IR method is readily satisfied. Under reasonable assumptions, the IR method for optimal control problems is shown to converge to a solution of the discretized problem. Convergence of a solution of the discretized problem to a solution of the continuous-time problem is also shown. It turns out that optimality phase equations of the IR method emanate from an associated Hamiltonian system, and so general Runge-Kutta discretization induces a symplectic partitioned Runge-Kutta scheme. A computational algorithm is described, and numerical experiments are made to demonstrate the working of the method for optimal control of the van der Pol system, employing the three-stage (order 6) Gauss-Legendre discretization.

Heroin users in Australia: population trends.

The aim of this paper is to identify certain important population trends among heroin users in Australia for the period 1971 - 97, such as: population growth, initiation, i.e. the number who were initiated to heroin in a given year, and quitting, i.e. the number that quit using heroin. For this purpose, we summarize and extract relevant characteristics from data from National Drug Strategy Household Survey (NDSHS 1998) conducted in Australia in 1998. We devise a systematic procedure to estimate historical trends from questions concerning past events. It is observed from our findings that the size of the heroin user population in Australia is in a sharp increase, especially from the early 1980s onwards. The general trend obtained for the period 1971 - 97 is strikingly similar to that obtained by Hall et al. (2000) for the dependent heroin user population in Australia, even though their study was based on different datasets and a different methodology. In our reconstruction of the time history we also detect a levelling-off prior to 1990. Initiation is also observed to be on a sharp increase. The latter trend is accompanied by a similar trend of quitting, perhaps indicating a relatively short heroin use career. A sharp decrease in both initiation and quitting is observed after 1990. In conclusion, in the case of the trend in the population of heroin users a high rate of growth has been identified that is consistent with the existing literature. In the process, we demonstrated that even a static survey such as NDSHS 1998 can, sometimes, be used to extract historical (dynamic) trends of certain important variables.

A new scalarization and numerical method for constructing the weak Pareto front of multi-objective optimization problems

A numerical technique is presented for constructing an approximation of the weak Pareto front of nonconvex multi-objective optimization problems, based on a new Tchebychev-type scalarization and its equivalent representations. First, existing results on the standard Tchebychev scalarization, the weak Pareto and Pareto minima, as well as the uniqueness of the optimal value in the Pareto front, are recalled and discussed for the case when the set of weak Pareto minima is the same as the set of Pareto minima, namely, when weak Pareto minima are also Pareto minima. Of the two algorithms we present, Algorithm 1 is based on this discussion. Algorithm 2, on the other hand, is based on the new scalarization incorporating rays associated with the weights of the scalarization in the value (or objective) space, as constraints. We prove two relevant results for the new scalarization. The new scalarization and the resulting Algorithm 2 are particularly effective in constructing an approximation of the weak Pareto sections of the front. We illustrate the working and capability of both algorithms by means of smooth and nonsmooth test problems with connected and disconnected Pareto fronts.

Optimization Over the Efficient Set of Multi-objective Convex Optimal Control Problems

We consider multi-objective convex optimal control problems. First we state a relationship between the (weakly or properly) efficient set of the multi-objective problem and the solution of the problem scalarized via a convex combination of objectives through a vector of parameters (or weights). Then we establish that (i) the solution of the scalarized (parametric) problem for any given parameter vector is unique and (weakly or properly) efficient and (ii) for each solution in the (weakly or properly) efficient set, there exists at least one corresponding parameter vector for the scalarized problem yielding the same solution. Therefore the set of all parametric solutions (obtained by solving the scalarized problem) is equal to the efficient set. Next we consider an additional objective over the efficient set. Based on the main result, the new objective can instead be considered over the (parametric) solution set of the scalarized problem. For the purpose of constructing numerical methods, we point to existing solution differentiability results for parametric optimal control problems. We propose numerical methods and give an example application to illustrate our approach.

Inexact Restoration for Euler Discretization of Box-Constrained Optimal Control Problems

The Inexact Restoration method for Euler discretization of state and control constrained optimal control problems is studied. Convergence of the discretized (finite-dimensional optimization) problem to an approximate solution using the Inexact Restoration method and convergence of the approximate solution to a continuous-time solution of the original problem are established. It is proved that a sufficient condition for convergence of the Inexact Restoration method is guaranteed to hold for the constrained optimal control problem. Numerical experiments employing the modelling language AMPL and optimization software Ipopt are carried out to illustrate the robustness of the Inexact Restoration method by means of two computationally challenging optimal control problems, one involving a container crane and the other a free-flying robot. The experiments interestingly demonstrate that one might be better-off using Ipopt as part of the Inexact Restoration method (in its subproblems) rather than using Ipopt directly on its own.

COMPUTATIONS FOR TIME-OPTIMAL BANG–BANG CONTROL USING A LAGRANGIAN FORMULATION

Abstract In this paper an algorithm is proposed to solve the problem of time-optimal bang–bang control of nonlinear systems from a given initial state to a given terminal state. The problem is reduced to the problem of minimising a Lagrangian subject to an equality constraint defined by the terminal state. Then a solution is obtained by solving a system of nonlinear equations. Examples are given so as to illustrate the algorithm presented.

A Deflected Subgradient Method Using a General Augmented Lagrangian Dualitywith Implications on Penalty Methods

We propose a duality scheme for solving constrained nonsmooth and nonconvex optimization problems. Our approach is to use a new variant of the deflected subgradient method for solving the dual problem. Our augmented Lagrangian function induces a primal–dual method with strong duality, that is, with zero duality gap. We prove that our method converges to a dual solution if and only if a dual solution exists. We also prove that all accumulation points of an auxiliary primal sequence are primal solutions. Our results apply, in particular, to classical penalty methods, since the penalty functions associated with these methods can be recovered as a special case of our augmented Lagrangians. Besides the classical augmenting terms given by the l 1- or l 2-norm forms, terms of many other forms can be used in our Lagrangian function. Using a practical selection of the step-size parameters, as well as various choices of the augmenting term, we demonstrate the method on test problems. Our numerical experiments indicate that it is more favourable to use an augmenting term of an exponential form rather than the classical l 1- or l 2-norm forms.

Time-optimal switching control for the US cocaine epidemic

Abstract Behrens et al. developed a control model for the cocaine epidemic in the US. They investigated the problem of allocating resources to treatment, where the objective is to minimize social and control costs. The optimal control strategy they found for a particular case of constrained budget resembles bang–bang control: a treatment-only period follows a prevention-only period. This motivated us to look at the problem of bringing the size of the epidemic down to a target level in minimum possible time, find the necessary number of switchings between treatment and prevention, and the instants these switchings occur. We carry out a computational analysis for the so-called early- and late-action scenarios, where the controls take effect from years 1967 and 1990, respectively. We investigate the problem in terms of the minimized process time, cost of control, and target ratio of heavy users. The best strategy in either scenario emerges as “first prevention, then treatment” with a rather surprising additional requirement: to reach the target in minimum possible time, the ratio of heavy users at the target should be prescribed as small as possible. This also yields the lowest cost. Further comparisons and interpretations are given.

Finding Interpolating Curves Minimizing

We study the problem of finding an interpolating curve passing through prescribed points in the Euclidean space. The interpolating curve minimizes the pointwise maximum length, i.e.,