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Future challenges for variational analysis
by
Jonathan Borwein
Newcastle and Dalhousie
Modern nonsmooth analysis is now roughly thirty-five years old. In this talk I shall briefly assess where the subject stands today from the perspective of both theory and applications. I will also discuss some open problems and current challenges for the subject.
Date received: November 2, 2008
Outer approximation schemes for generalized semi-infinite variational inequality problems
by
Regina Sandra Burachik
University of South Australia
Coauthors: Lopes, J. O. (Universidade Federal do Piaui, Brazil)
We introduce and analyze outer approximation schemes for solving variational inequality problems in which the constraint set is as in generalized semi-infinite programming. We call these problems Generalized Semi-Infinite Variational Inequality Problems. First, we establish convergence results of our method under standard boundedness assumptions. Second, we use suitable Tikhonov-like regularizations for establishing convergence in the unbounded case.
Date received: October 23, 2008
Direction-set updates for derivative-free optimization
by
Ian Coope
University of Canterbury
The updating of direction sets in direct search methods for unconstrained optimization is examined. Both weak and strong quasi-Newton updates are considered together with other simple quadratic interpolation conditions. Efficient and numerically stable techniques are described for implementing the appropriate updates. The updating schemes are applicable to both line search and trust region algorithms as well as some newer grid-based methods for derivative-free optimization and the updates considered can usually be calculated in O(n2) arithmetic operations.
Date received: November 2, 2008
Kinds of vector invex
by
Bruce Craven
University of Melbourne
Necessary Lagrangian conditions for a constrained minimum become come sufficient under generalized convex assumptions, in particular invex, and duality results follow. Many classes of vector functions with properties related to invex have been studied, but it has not been clear how far these classes are distinct. Various inclusions between these classes are now established. Some modifications of invex can be regarded as perturbations of invex. There is a stability criterion for when the invex property is preserved under small perturbations. Some results extend to nondifferentiable (Lipschitz) functions.
Date received: October 27, 2008
A new least squares best fit problem for utility estimation with application to the fitting of elasticities to data in CGE modelling
by
Andrew Eberhard
RMIT University
Coauthors: A. Eberhard, S. Schreider, L. Stojkov and D. Ralph
We consider the problem of fitting of a utility to a finite sample of demand data. Theory is developed to justify this process which shows that this process provides a partial positive answer to the problem of revealed preference when no sampling errors are present. It is showed that even when the underlying utility is not consistent with a concavity assumption one can still form approximations involving concave utilities. Conditions are given in terms of the boundedness of parameters fitted in the approximate Afriat utilities that ensure the approximations converge to a concave utility. When sampling errors are present one must solve a nonlinear best fit problem of unique character. Application is then made to the estimation of elasticities that are used in economic models. Some numerical simulations are provided.
Date received: October 29, 2008
A multistage stochastic programming approach to open pit mine production scheduling with uncertain geology
by
Gary Froyland
University of New South Wales
Coauthors: Natashia Boland, University of Newcastle;
Irina Dumitrescu, University of New South Wales
The Open Pit Mine Production Scheduling Problem (OPMPSP) studied in recent years is usually based on a single geological estimate of material to be excavated and processed over a number of decades. However techniques have now been developed to generate multiple stochastic geological estimates that more accurately describe the uncertain geology. While some attempts have been made to use such multiple estimates in mine production scheduling, none of these allow mining and processing decisions to flexibly adapt over time, in response to observation of the geological properties of the material mined. In this paper, we use multiple geological estimates in a mixed integer multistage stochastic programming approach, in which decisions made in later time periods can depend on observations of the geological properties of the material mined in earlier periods. Since the material mined in earlier periods is determined by our decisions, the information received about uncertain properties, and when that information is available, is decision-dependent. Thus we tackle the difficult case of stochastic programming with endogeneous uncertainty. We extend a successful mixed integer programming formulation of the OPMPSP to this stochastic case, and show that non-anticipativity can be modelled with linear constraints involving variables already present in the model. We extend this observation to the general class of endogenous stochastic programs, and exploit the special structure of our model to show that in some cases we can omit a significant proportion of these constraints. Using data supplied by our industry partner, (a multinational mining company), we show that this approach is reasonably tractable, and demonstrate the improvements that can be made to mine schedules through the explicit use of multiple geological estimates.
Date received: October 27, 2008
Necessary and sufficient conditions for inversion of perturbed linear operators on Banach space.
by
Phil Howlett
University of South Australia
Coauthors: Amie Albrecht, Charles Pearce
In this paper we find necessary and sufficient conditions for the existence of a Laurent series expansion with a finite order pole at the origin for a perturbed bounded linear operator on Banach space.
Date received: October 30, 2008
Runge-Kutta discretization and inexact restoration for optimal control
by
C Yalcin Kaya
University of South Australia
Coauthors: Bulent Karasozen
A computational technique for a class of optimal control problems is presented. First Runge-Kutta discretization is carried out to obtain a finite-dimensional approximation of the continuous-time problem. Then an inexact restoration (IR) method is applied to the discretized problem to find an approximate solution. Convergence of the technique to a solution of the continuous-time problem is facilitated, under some general conditions, by convergence of the IR method and convergence of the discrete (approximate) solution as finer subdivisions are taken. Numerical experiments are presented for a discussion of the technique.
Date received: October 29, 2008
SIP approach to continuously constrained LQ optimal control problems via piecewise polynomial control parameterization
by
Yanqun Liu
Department of Mathematics and Statistics, RMIT University
In this paper, we consider the class of LQ optimal control problems with continuous constraints involving both the state and control. We extend an existing SIP method for the cases where the constraints involves only the system states. The existing method employs piecewise constant control to reduce the optimal control problem to an SIP problem. Here we use piecewise polynomial control instead. We provide convergence results with treatment of the control term in the constraint function. We present a number of illustrative examples demonstrating an improved accuracy over piecewise constant controls without increasing the size of the resulted SIP problem.
Date received: October 24, 2008
Necessary optimality conditions for some control problems of elliptic equations with venttsel boundary conditions
by
Yousong Luo
RMIT University
In this paper we derive a necessary optimality condition for a local optimal solution of some control problems. These optimal control problems are governed by a semi-linear Vettsel boundary value problem of a linear elliptic equation. The control is applied to the state equation via the boundary of the domain and a functional of the control together with the solution of the state equation under such a control will be minimized. A constraint on the solution of the state equation is also considered.
Date received: October 23, 2008
A stochastic direct method for bound constrained non-smooth global optimization
by
Chris Price
Maths and Stats, University of Canterbury
Coauthors: M. Reale and B. L. Robertson
A stochastic algorithm for global optimization subject to simple bounds is described. The algorithm is in the spirit of the direct algorithm of Jones, Perttunen, and Stuckman. Like direct it generates succcessively finer covers of the feasible region. Each cover consists of a finite number of boxes, where each box is defined by simple bounds on each variable. Its principal differences are that it subdivides each box into two rather than three smaller boxes, and that it calculates the objective function at a randomly selected point in each box, rather than the box's centre. The stochastic nature of the method permits a limited memory version to be developed. The sequence of best known function values is shown to converge to the essential global minimum with probability one on non-smooth functions. Numerical results are presented.
Date received: October 27, 2008
Optimal attitude control of an accompanying satellite rotating around the space station
by
Fang Wang
RMIT University
Coauthors: Pavel M. Trivailo,Honghua Zhang
This paper deals with the optimal control of an accompanying satellite rotating around the space station in the presence of sinusoidal disturbances. The concept of accompanying satellites (AS) around the space station (SS) is introduced. Both the AS and the SS are modeled as rigid bodies with the reference coordinate frames described for the AS pointing to the SS. There are many functions for the AS of the SS such as navigation, relaying communication data and inspecting the SS. Since all of these functions require attitude control of the AS, it is necessary to study the optimal control for the AS rotating around the SS.
Untill now there have been many studies about the optimal control of satellites. However, all of them did not consider the case when the satellite is underactuated, i.e., actuators in one or two dimensions are failed, though the fact is that the AS has a probability of disabled (or damaged or malfunctioned) actuators during its running period. Hence it is also necessary to study the problem when AS is underactuated, which is the novelty of the paper. Without loss of generality, the underactuated axis is assumed to be the third axis of the AS.
The purpose of this paper can be stated as designing an optimal control law so that the underactuated AS can achieve suitable attitude in accordance with the expected thrust direction before orbit maneuvering, and then attain reorientation towards the desired direction (e.g., the SS) after the orbit maneuvering with the least thrust. Based on this purpose, the paper first defines the reference coordinate, frames for the AS pointing to the SS according to different missions. Then by using unit quaternion, the full set of nonlinear equations of motion is derived. These equations are solved numerically using direct transcription method to obtain optimal solution for the underactuated satellite. The direct method seeks to transform the continuous optimal control problem into a discrete mathematical programming problem, which in turn is solved using a non-linear programming algorithm. By discretizing the state and control variables at a series of nodes, the integration of the dynamical equations of motion is not required. The state equations are enforced as constraints by using interpolating polynomials and an implicit integration scheme is used in each discrete segment, which ensures fast computational times. The Chebyshev-pseudospectral method, due to its ease of implementation, high accuracy and fast computation times, was chosen as the direct optimization method to be employed to solve the problem. The analytical and simulation results show that the proposed control law is effective in the case of failed actuators.
Further research in this area is discussed. It involves cases when the AS has flexible attachments; the AS is under disturbances; and the inertia matrix is unknown or poorly known, of which the controller for the underactuated AS will become more complicated.
Date received: September 28, 2008
Overshoot characterisation for continuous-time systems
by
Rob Wenczel
RMIT University
Coauthors: Robin Hill (RMIT)
This talk is concerned with the design of feedback controllers to optimally track a step input. For this problem the amount of overshoot in the system response is an issue of considerable engineering significance. For a given open-loop system, there are fundamental limitations on the extent to which a linear time-invariant controller can reduce the overshoot response to a step. Existing results for discrete-time systems use convex optimisation tools, with density arguments to achieve a tight bound.
We show that the same techniques yield results of identical structure, for the case of continuous-time systems
Date received: October 16, 2008
Solving large scale highly nonlinear systems of equations and spherical designs
by
Rob Womersley
School of Mathematics and Statistics, University of New South Wales
Spherical L-designs are sets of N points on the unit sphere such that the average of the function values at the points gives the integral over the sphere for any spherical polynomial of degree at most L. Thus they provide the nodes of an equal weight quadrature rule of precision L for the sphere.
Building on a new characterization of spherical designs by Sloan and Womersley, spherical designs can be characterized by finding a global minimum of zero for an objective function or solving a system of nonlinear equations. This talk concentrates on issues related to solving the resulting large scale (up to degree 100 with 10, 000 variables and equations) highly nonlinear system of equations and difficulties with being sure that there is a true solution near to the computed solution.
Date received: October 29, 2008