Development of self-adaptive back propagation and derivative free training algorithms in artificial neural networks

Three new iterative, dynamically self-adaptive, derivative-free and training parameter free artificial neural network (ANN) training algorithms are developed. They are defined as self-adaptive back propagation, multi-directional and restart ANN training algorithms. The descent direction in self-adaptive back propagation training is determined implicitly by a central difference approximation scheme, which chooses its step size according to the convergence behavior of the error function. This approach trains an ANN when the gradient information of the corresponding error function is not readily available. The self- adaptive variable learning rates per epoch are determined dynamically using a constrained interpolation search. As a result, appropriate descent to the error function is achieved. The multi-directional training algorithm is self-adaptive and derivative free. It orients an initial search vector in a descent location at the early stage of training. Individual learning rates and momentum term for all the ANN weights are determined optimally. The search directions are derived from rectilinear and Euclidean paths, which explore stiff ridges and valleys of the error surface to improve training. The restart training algorithm is derivative free. It redefines a de-generated simplex at a re-scale phase. This multi-parameter training algorithm updates ANN weights simultaneously instead of individually. The descent directions are derived from the centroid of a simplex along a reflection point opposite to the worst vertex. The algorithm is robust and has the ability to improve local search. These ANN training methods are appropriate when there is discontinuity in corresponding ANN error function or the Hessian matrix is ill conditioned or singular. The convergence properties of the algorithms are proved where possible. All the training algorithms successfully train exclusive OR (XOR), parity, character-recognition and forecasting problems. The simulation results with XOR, parity and character recognition problems suggest that all the training algorithms improve significantly over the standard back propagation algorithm in average number of epoch, function evaluations and terminal function values. The multivariate ANN calibration problem as a regression model with small data set is relatively difficult to train. In forecasting problems, an ANN is trained to extrapolate the data in validation period. The extrapolation results are compared with the actual data. The trained ANN performs better than the statistical regression method in mean absolute deviations; mean squared errors and relative percentage error. The restart training algorithm succeeds in training a problem, where other training algorithms face difficulty. It is shown that a seasonal time series problem possesses a Hessian matrix that has a high condition number. Convergence difficulties as well as slow training are therefore not atypical. The research exploits the geometry of the error surface to identify self-adaptive optimized learning rates and momentum terms. Consequently, the algorithms converge with high success rate. These attributes brand the training algorithms as self-adaptive, automatic, parameter free, efficient and easy to use

Indefinite Knapsack Separable Quadratic Programming: Methods and Applications

Quadratic programming (QP) has received significant consideration due to an extensive list of applications. Although polynomial time algorithms for the convex case have been developed, the solution of large scale QPs is challenging due to the computer memory and speed limitations. Moreover, if the QP is nonconvex or includes integer variables, the problem is NP-hard. Therefore, no known algorithm can solve such QPs efficiently. Alternatively, row-aggregation and diagonalization techniques have been developed to solve QP by a sub-problem, knapsack separable QP (KSQP), which has a separable objective function and is constrained by a single knapsack linear constraint and box constraints. KSQP can therefore be considered as a fundamental building-block to solve the general QP and is an important class of problems for research. For the convex KSQP, linear time algorithms are available. However, if some quadratic terms or even only one term is negative in KSQP, the problem is known to be NP-hard, i.e. it is notoriously difficult to solve. The main objective of this dissertation is to develop efficient algorithms to solve general KSQP. Thus, the contributions of this dissertation are five-fold. First, this dissertation includes comprehensive literature review for convex and nonconvex KSQP by considering their computational efficiencies and theoretical complexities. Second, a new algorithm with quadratic time worst-case complexity is developed to globally solve the nonconvex KSQP, having open box constraints. Third, the latter global solver is utilized to develop a new bounding algorithm for general KSQP. Fourth, another new algorithm is developed to find a bound for general KSQP in linear time complexity. Fifth, a list of comprehensive applications for convex KSQP is introduced, and direct applications of indefinite KSQP are described and tested with our newly developed methods. Experiments are conducted to compare the performance of the developed algorithms with that of local, global, and commercial solvers such as IBM CPLEX using randomly generated problems in the context of certain applications. The experimental results show that our proposed methods are superior in speed as well as in the quality of solutions