UcSolve
Purpose
Solve unconstrained nonlinear optimization problems with simple bounds on the variables.
ucSolve solves problems of the form
where Failed to parse (unknown function "\MATHSET"): {\displaystyle x,x_{L},x_{U}\in \MATHSET{R} ^{n}}
.
Calling Syntax
Result = ucSolve(Prob, varargin)
Inputs
Prob | Problem description structure. The following fields are used: | |
x_L | Lower bounds on the variables. | |
x_U | Upper bounds on the variables. | |
x_0 | Starting point. | |
FUNCS.f | Name of m-file computing the objective function f (x). | |
FUNCS.g | Name of m-file computing the gradient vector g(x). | |
FUNCS.H | Name of m-file computing the Hessian matrix H (x). | |
f_Low | Lower bound on function value. | |
Solver.Alg | Solver algorithm to be run:
0: Gives default, either Newton or BFGS. 1: Newton with subspace minimization, using SVD. 2: Safeguarded BFGS with inverse Hessian updates (standard). 3: Safeguarded BFGS with Hessian updates. 4: Safeguarded DFP with inverse Hessian updates. 5: Safeguarded DFP with Hessian updates. 6: Fletcher-Reeves CG. 7: Polak-Ribiere CG. 8: Fletcher conjugate descent CG-method. | |
Solver.Method | Method used to solve equation system:
0: SVD (default). 1: LU-decomposition. 2: LU-decomposition with pivoting. 3: Matlab built in QR. 4: Matlab inversion. 5: Explicit inverse. | |
Solver.Method | Restart or not for C-G method:
0: Use restart in CG-method each n:th step. 1: Use restart in CG-method each n:th step. | |
LineParam | Structure with line search parameters, see routine LineSearch and TOMLAB Appendix A. | |
optParam | Structure with special fields for optimization parameters, see TOMLAB Appendix A
Fields used are: eps_absf, eps_f, eps_g, eps_x, eps_Rank, MaxIter, wait, size_x, xTol, size_f, LineSearch, LineAlg, xTol, IterPrint and QN InitMatrix. | |
PriLevOpt | Print level. | |
varargin | Other parameters directly sent to low level routines. |
Outputs
Result | Structure with result from optimization. The following fields are changed: | |
---|---|---|
x_k | Optimal point. | |
f_k | Function value at optimum. | |
g_k | Gradient value at optimum. | |
H_k | Hessian value at optimum. | |
B_k | Quasi-Newton approximation of the Hessian at optimum. | |
v_k | Lagrange multipliers. | |
x_0 | Starting point. | |
f_0 | Function value at start. | |
xState | State of each variable, described in TOMLAB Appendix B. | |
Iter | Number of iterations. | |
ExitFlag | 0 if convergence to local min. Otherwise errors. | |
Inform | Binary code telling type of convergence:
1: Iteration points are close. 2: Projected gradient small. 4: Relative function value reduction low for LowIts iterations. 101: Maximum number of iterations reached. 102: Function value below given estimate. 104: Convergence to a saddle point. | |
Solver | Solver used. | |
SolverAlgorithm | Solver algorithm used. | |
Prob | Problem structure used. |
Description
The solver ucSolve includes several of the most popular search step methods for unconstrained optimization. The search step methods included in ucSolve are: the Newton method, the quasi-Newton BFGS and DFP methods, the Fletcher-Reeves and Polak-Ribiere conjugate-gradient method, and the Fletcher conjugate descent method. The quasi-Newton methods may either update the inverse Hessian (standard) or the Hessian itself. The Newton method and the quasi-Newton methods updating the Hessian are using a subspace minimization technique to handle rank problems. The quasi-Newton algorithms also use safe guarding techniques to avoid rank problem in the updated matrix. The line search algorithm in the routine LineSearch is a modified version of an algorithm by Fletcher. Bound constraints are treated as described in Gill, Murray and Wright. The accuracy in the line search is critical for the performance of quasi-Newton BFGS and DFP methods and for the CG methods. If the accuracy parameter Prob.LineParam.sigma is set to the default value 0.9, ucSolve changes it automatically according to:
Prob.Solver.Alg | Prob.LineParam.sigma |
---|---|
4,5 (DFP) | 0.2 |
6,7,8 (CG) | 0.01 |
M-files Used
ResultDef.m, LineSearch.m, iniSolve.m, tomSolve.m, endSolve.m