TOMLAB OPTIMIZATION ENVIRONMENT: ucSolve

ucSolve

Purpose

For solving unconstrained nonlinear optimization problems with simple bounds on the variables.

Syntax

   Result = ucSolve(Prob, varargin);

Description

ucSolve is a solver implementing several algorithms for unconstrained minimization with bound constraints,

ucSolve implements the following algorithms:

Algorithm:

• 0 = Default algorithm (BFGS or Newton)
• 1 = Newton with subspace minimization, using SVD
• 2 = Safeguarded BFGS with standard inverse Hessian update
• 3 = Safeguarded BFGS with Hessian update, and SVD or LU to solve
• 4 = Safeguarded DFP with standard inverse Hessian update
• 5 = Safeguarded DFP with Hessian update, and SVD or LU to solve
• 6 = Fletcher-Reeves CG
• 7 = Polak-Ribiere CG
• 8 = Fletcher conjugate descent CG-method

Set the field Prob.Solver.Alg to one of the numbers above

Note: The accuracy in the line search is critical for the performance if quasi-Newton BFGS and DFP methods, as well as the CG methods. If the accuracy parameter, Prob.LineParam.sigma == 0.9 (standard default), it is changed by ucSolve to:

       Prob.Solver.Alg == 4,5   => sigma =  0.2
       Prob.Solver.Alg == 6,7,8 => sigma =  0.01

Bound constraints treated as described in Gill, Murray, Wright: Practical Optimization, Academic Press, 1981.

Solver.Method, method to solve equation system (Solver.Alg in [0,5])

• 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: (Solver.Alg in [6,8])

• 0 = Use restart in CG-method each n:th step.
• 1 = Use restart in CG-method each n:th step.

Problem

    min      f(x)
	
        x_L <= x <= x_U

Input Parameters

Use tomMenu.m or tomGUI.m for interactive run

Use conAssign.m (or probAssign) to initialize the Prob structure in the TOMLAB Quick format. Fields used in structure Prob:

    USER.f    The routine to compute the function value, given as a string
    USER.g    The routine to compute the gradient vector, given as a string
              If empty, or Prob.NumDiff ~=0 numerical differences are used
              If Prob.AutoDiff > 0, automatic differentiation is used
    USER.H    The routine to compute the Hessian, given as a string
              Only used for Newtons method.
              If empty, or Prob.NumDiff < 0 numerical differences are used
              If Prob.AutoDiff > 0, automatic differentiation is used
    x_0       Starting point
    x_L       Lower bounds for x
    x_U       Upper bounds for x
 PriLevOpt    Print level

    Solver.Alg    Algorithm number
    Solver.Method Search direction solution technique

    optParam   Structure with optimization parameters
       eps_absf  Absolute convergence tol on function value
       eps_f     Relative change in f, convergence tolerance     
                 ucSolve will stop if the test is true for LowIts iterations
                 LowIts = 10; now, to change it, change on line 182.
       eps_g     Gradient convergence tolerance
       eps_Rank  Rank test tolerance
       eps_x     Convergence tolerance in x
       MaxIter   Maximal number of iterations
       optParam.QN_InitMatrix  Initial Quasi-Newton matrix, empty ==> eye(n),
       IterPrint Print short information each iteration
       size_x    Approximate size of optimal variable values, normally 1
       size_f    Approximate size of optimal function value, normally 1 
       xTol      Tolerance to judge if x is close to bounds   
                               otherwise use identity matrix
     LineParam   Line search parameters, see LineSearch.m

  PriLev:   Print level:  0 None, 1 Final result, 2 Each iteration, short
            3 Each iteration, more info, 4 Line search info and Hessian

The Prob structure could be created in the TOMLAB Quick format with calls to conAssign.m (or probAssign and mFiles), or in the Init File format, see Users Guide.

Extra Parameters

 VARARGIN: User defined parameters passed to f,g,H, r and J, and c and dc.

Output Parameters

 Result      Structure with results from optimization
    x_k      Optimal point
    v_k      Lagrange multipliers for bound constraints
    f_k      Function value at optimum
    g_k      Gradient vector at optimum
    x_0      Starting value vector
    H_k      The Hessian matrix, if computed (Newtons method) 
    B_k      The Quasi Newton matrix
    xState   Variable: Free==0; On lower == 1; On upper == 2; Fixed == 3;
    Iter     Number of iterations
    ExitFlag Flag giving exit status
    ExitTest Text string giving ExitFlag and Inform information
    Inform   Code telling type of convergence
             1   Iteration points are close.
             2   Projected gradient small.
             3   Iteration points are close and projected gradient small.
             4   Relative function value reduction low for 10 iterations.
             5   Iteration points are close and relative function value
                 reduction low for 10 iterations.
             6   Projected gradient small and relative function value
                 reduction low for 10 iterations.
             7   Iteration points are close, projected gradient small and
                 relative function value reduction low for 10 iterations.
           101   Max no of iterations reached.
           102   Function value below given estimate.
           103   Convergence to saddle point (eigenvalues computed)

Tnnls