A new recurslve method for the analysis of linear time invariant dynamic systems via double-term triangular functions （DTTF） in state space environment

This paper presents a new recursive method for system analysis via double-term triangular functions (DTTF) in state space environment. The proposed method uses orthogonal triangular function sets and proves to be more accurate as compared to single term Walsh series (STWS) method with respect to mean integral square error (MISE). This has been established theoretically and comparison of error with respect to MISE is presented for clarity. A numerical example is treated to establish the proposed method. Relevant curves for the solutions of states of the dynamic system are also presented with plots of percentage error for DTTF-based analysis.     

A new numerical method for pricing fixed-rate mortgages with prepayment and default options

In this paper we consider the valuation of fixed-rate mortgages including prepayment and default options, where the underlying stochastic factors are the house price and the interest rate. The mathematical model to obtain the value of the contract is posed as a free boundary problem associated to a partial differential equation (PDE) model. The equilibrium contract rate is determined by using an iterative process. Moreover, appropriate numerical methods based on a Lagrange–Galerkin discretization of the PDE, an augmented Lagrangian active set method and a Newton iteration scheme are proposed. Finally, some numerical results to illustrate the performance of the numerical schemes, as well as the qualitative and quantitative behaviour of solution and the optimal prepayment boundary are presented.     

An augmented Lagrangian approach to non-convex SAO using diagonal quadratic approximations

Successful gradient-based sequential approximate optimization (SAO) algorithms in simulation-based optimization typically use convex separable approximations. Convex approximations may however not be very efficient if the true objective function and/or the constraints are concave. Using diagonal quadratic approximations, we show that non-convex approximations may indeed require significantly fewer iterations than their convex counterparts. The nonconvex subproblems are solved using an augmented Lagrangian (AL) strategy, rather than the Falk-dual, which is the norm in SAO based on convex subproblems. The results suggest that transformation of large-scale optimization problems with only a few constraints to a dual form via convexification need sometimes not be required, since this may equally well be done using an AL formulation.     

An augmented Lagrangian relaxation for analytical target cascading using the alternating direction method of multipliers

Analytical target cascading is a method for design optimization of hierarchical, multilevel systems. A quadratic penalty relaxation of the system consistency constraints is used to ensure subproblem feasibility. A typical nested solution strategy consists of inner and outer loops. In the inner loop, the coupled subproblems are solved iteratively with fixed penalty weights. After convergence of the inner loop, the outer loop updates the penalty weights. The article presents an augmented Lagrangian relaxation that reduces the computational cost associated with ill-conditioning of subproblems in the inner loop. The alternating direction method of multipliers is used to update penalty parameters after a single inner loop iteration, so that subproblems need to be solved only once. Experiments with four examples show that computational costs are decreased by orders of magnitude ranging between 10 and 1000.     

Optimization of dynamic response using a monolithic-time formulation

The design of systems for dynamic response may involve constraints that need to be satisfied over an entire time interval or objective functions evaluated over the interval. Efficiently performing the constrained optimization is challenging, since the typical response is implicitly linked to the design variables through a numerical integration of the governing differential equations. Evaluating constraints is costly, as is the determination of sensitivities to variations in the design variables. In this paper, we investigate the application of a temporal spectral element method to the optimization of transient and time-periodic responses of fundamental engineering systems. Through the spectral discretization, the response is computed globally, thereby enabling a more explicit connection between the response and design variables and facilitating the efficient computation of response sensitivities. Furthermore, the response is captured in a higher order manner to increase analysis accuracy. Two applications of the coupling of dynamic response optimization with the temporal spectral element method are demonstrated. The first application, a one-degree-of-freedom, linear, impact absorber, is selected from the auto industry, and tests the ability of the method to treat transient constraints over a large-time interval. The second application, a related mass-spring-damper system, shows how the method can be used to obtain work and amplitude optimal time-periodic control force subject to constraints over a periodic time interval. This research was performed while the first author held a National Research Council Research Associateship Award at the Air Force Research Laboratory. An early version of this paper was presented at the 46th AIAA Aerospace Sciences Meeting and Exhibit, Jan 7–10, 2008, Reno, Nevada.     

The second generation of self-organizing adaptive penalty strategy for constrained genetic search

Penalty function approaches have been extensively applied to genetic algorithms for tackling constrained optimization problems. The effectiveness of the genetic searches to locate the global optimum on constrained optimization problems often relies on the proper selections of many parameters involved in the penalty function strategies. A successful genetic search is often completed after a number of genetic searches with varied combinations of penalty function related parameters. In order to provide a robust and effective penalty function strategy with which the design engineers use genetic algorithms to seek the optimum without the time-consuming tuning process, the self-organizing adaptive penalty strategy (SOAPS) for constrained genetic searches was proposed. This paper proposes the second generation of the self-organizing adaptive penalty strategy (SOAPS-II) to further improve the effectiveness and efficiency of the genetic searches on constrained optimization problems, especially when equality constraints are involved. The results of a number of illustrative testing problems show that the SOAPS-II consistently outperforms other penalty function approaches.