12 OptimizationThe contents of this section currently describes deprecated classes. Please refer to the new API description. 12.1 OverviewThe optimization package provides algorithms to optimize (i.e. either minimize or maximize) some objective or cost function. The package is split in several sub-packages dedicated to different kind of functions or algorithms.
The top level optimization package provides common interfaces for the optimization algorithms provided in sub-packages. The main interfaces defines defines optimizers and convergence checkers. The functions that are optimized by the algorithms provided by this package and its sub-packages are a subset of the one defined in the analysis package, namely the real and vector valued functions. These functions are called objective function here. When the goal is to minimize, the functions are often called cost function, this name is not used in this package. The type of goal, i.e. minimization or maximization, is defined by the enumerated GoalType which has only two values: MAXIMIZE and MINIMIZE. Optimizers are the algorithms that will either minimize or maximize, the objective function by changing its input variables set until an optimal set is found. There are only four interfaces defining the common behavior of optimizers, one for each supported type of objective function:
Despite there are only four types of supported optimizers, it is possible to optimize a transform a non-differentiable multivariate vectorial function by converting it to a non-differentiable multivariate real function thanks to the LeastSquaresConverter helper class. The transformed function can be optimized using any implementation of the MultivariateOptimizer interface. For each of the four types of supported optimizers, there is a special implementation which wraps a classical optimizer in order to add it a multi-start feature. This feature call the underlying optimizer several times in sequence with different starting points and returns the best optimum found or all optima if desired. This is a classical way to prevent being trapped into a local extremum when looking for a global one. 12.2 Univariate FunctionsA UnivariateOptimizer is used to find the minimal values of a univariate real-valued function f. These algorithms usage is very similar to root-finding algorithms usage explained in the analysis package. The main difference is that the solve methods in root finding algorithms is replaced by optimize methods. 12.3 Linear ProgrammingThis package provides an implementation of George Dantzig's simplex algorithm for solving linear optimization problems with linear equality and inequality constraints. 12.4 Direct MethodsDirect search methods only use cost function values, they don't need derivatives and don't either try to compute approximation of the derivatives. According to a 1996 paper by Margaret H. Wright (Direct Search Methods: Once Scorned, Now Respectable), they are used when either the computation of the derivative is impossible (noisy functions, unpredictable discontinuities) or difficult (complexity, computation cost). In the first cases, rather than an optimum, a not too bad point is desired. In the latter cases, an optimum is desired but cannot be reasonably found. In all cases direct search methods can be useful. Simplex-based direct search methods are based on comparison of the cost function values at the vertices of a simplex (which is a set of n+1 points in dimension n) that is updated by the algorithms steps. The instances can be built either in single-start or in multi-start mode. Multi-start is a traditional way to try to avoid being trapped in a local minimum and miss the global minimum of a function. It can also be used to verify the convergence of an algorithm. In multi-start mode, the minimizesmethod returns the best minimum found after all starts, and the etMinima method can be used to retrieve all minima from all starts (including the one already provided by the minimizes method). The direct package provides four solvers:
The first two simplex-based methods do not handle simple bounds constraints by themselves. However there are two adapters( MultivariateFunctionMappingAdapter and MultivariateFunctionPenaltyAdapter) that can be used to wrap the user function in such a way the wrapped function is unbounded and can be used with these optimizers, despite the fact the underlying function is still bounded and will be called only with feasible points that fulfill the constraints. Note however that using these adapters are only a poor man solutions to simple bounds optimization constraints. Better solutions are to use an optimizer that directly supports simple bounds. Some caveats of the mapping adapter solution are that
The last methods do handle simple bounds constraints directly, so the adapters are not needed with them. 12.5 General CaseThe general package deals with non-linear vectorial optimization problems when the partial derivatives of the objective function are available. One important class of estimation problems is weighted least squares problems. They basically consist in finding the values for some parameters p_{k} such that a cost function J = sum(w_{i}(mes_{i} - mod_{i})^{2}) is minimized. The various (target_{i} - model_{i}(p_{k})) terms are called residuals. They represent the deviation between a set of target values target_{i} and theoretical values computed from models model_{i} depending on free parameters p_{k}. The w_{i} factors are weights. One classical use case is when the target values are experimental observations or measurements. Solving a least-squares problem is finding the free parameters p_{k} of the theoretical models such that they are close to the target values, i.e. when the residual are small. Two optimizers are available in the general package, both devoted to least-squares problems. The first one is based on the Gauss-Newton method. The second one is the Levenberg-Marquardt method. In order to solve a vectorial optimization problem, the user must provide it as an object implementing the DifferentiableMultivariateVectorFunction interface. The object will be provided to the estimate method of the optimizer, along with the target and weight arrays, thus allowing the optimizer to compute the residuals at will. The last parameter to the estimate method is the point from which the optimizer will start its search for the optimal point.
In addition to least squares solving, the NonLinearConjugateGradientOptimizer class provides a non-linear conjugate gradient algorithm to optimize DifferentiableMultivariateFunction. Both the Fletcher-Reeves and the Polak-Ribière search direction update methods are supported. It is also possible to set up a preconditioner or to change the line-search algorithm of the inner loop if desired (the default one is a Brent solver). The PowellOptimizer provides an optimization method for non-differentiable functions. |