Random Initialization and Multiple Restarts of the solnp solver.
When the objective function is non-smooth or has many local minima, it is hard to judge the optimality of the solution, and this usually depends critically on the starting parameters. This function enables the generation of a set of randomly chosen parameters from which to initialize multiple restarts of the solver (see note for details).
gosolnp(pars = NULL, fixed = NULL, fun, eqfun = NULL, eqB = NULL, ineqfun = NULL, ineqLB = NULL, ineqUB = NULL, LB = NULL, UB = NULL, control = list(), distr = rep(1, length(LB)), distr.opt = list(), n.restarts = 1, n.sim = 20000, cluster = NULL, rseed = NULL, ...)
pars: The starting parameter vector. This is not required unless the fixed option is also used.
fixed: The numeric index which indicates those parameters which should stay fixed instead of being randomly generated.
fun: The main function which takes as first argument the parameter vector and returns a single value.
eqfun: (Optional) The equality constraint function returning the vector of evaluated equality constraints.
eqB: (Optional) The equality constraints.
ineqfun: (Optional) The inequality constraint function returning the vector of evaluated inequality constraints.
ineqLB: (Optional) The lower bound of the inequality constraints.
ineqUB: (Optional) The upper bound of the inequality constraints.
LB: The lower bound on the parameters. This is not optional in this function.
UB: The upper bound on the parameters. This is not optional in this function.
control: (Optional) The control list of optimization parameters. The eval.type
option in this control list denotes whether to evaluate the function as is and exclude inequality violations in the final ranking (default, value = 1), else whether to evaluate a penalty barrier function comprised of the objective and all constraints (value = 2). See solnp function documentation for details of the remaining control options.
distr: A numeric vector of length equal to the number of parameters, indicating the choice of distribution to use for the random parameter generation. Choices are uniform (1), truncated normal (2), and normal (3).
distr.opt: If any choice in distr was anything other than uniform (1), this is a list equal to the length of the parameters with sub-components for the mean and sd, which are required in the truncated normal and normal distributions.
n.restarts: The number of solver restarts required.
n.sim: The number of random parameters to generate for every restart of the solver. Note that there will always be significant rejections if inequality bounds are present. Also, this choice should also be motivated by the width of the upper and lower bounds.
cluster: If you want to make use of parallel functionality, initialize and pass a cluster object from the parallel package (see details), and remember to terminate it!
rseed: (Optional) A seed to initiate the random number generator, else system time will be used.
...: (Optional) Additional parameters passed to the main, equality or inequality functions
Given a set of lower and upper bounds, the function generates, for those parameters not set as fixed, random values from one of the 3 chosen distributions. Depending on the eval.type option of the control
argument, the function is either directly evaluated for those points not violating any inequality constraints, or indirectly via a penalty barrier function jointly comprising the objective and constraints. The resulting values are then sorted, and the best N (N = random.restart) parameter vectors (corresponding to the best N objective function values) chosen in order to initialize the solver. Since version 1.14, it is up to the user to prepare and pass a cluster object from the parallel package for use with gosolnp, after which the parLapply function is used. If your function makes use of additional packages, or functions, then make sure to export them via the clusterExport
function of the parallel package. Additional arguments passed to the solver via the ... option are evaluated and exported by gosolnp to the cluster.
A list containing the following values: - pars: Optimal Parameters.
convergence: Indicates whether the solver has converged (0) or not (1).
values: Vector of function values during optimization with last one the value at the optimal.
lagrange: The vector of Lagrange multipliers.
hessian: The Hessian at the optimal solution.
ineqx0: The estimated optimal inequality vector of slack variables used for transforming the inequality into an equality constraint.
nfuneval: The number of function evaluations.
elapsed: Time taken to compute solution.
start.pars: The parameter vector used to start the solver
Y.Ye, Interior algorithms for linear, quadratic, and linearly constrained non linear programming, PhD Thesis, Department of EES Stanford University, Stanford CA.
Hu, X. and Shonkwiler, R. and Spruill, M.C. Random Restarts in Global Optimization, 1994, Georgia Institute of technology, Atlanta.
Alexios Ghalanos and Stefan Theussl
Y.Ye (original matlab version of solnp)
The choice of which distribution to use for randomly sampling the parameter space should be driven by the user's knowledge of the problem and confidence or lack thereof of the parameter distribution. The uniform distribution indicates a lack of confidence in the location or dispersion of the parameter, while the truncated normal indicates a more confident choice in both the location and dispersion. On the other hand, the normal indicates perhaps a lack of knowledge in the upper or lower bounds, but some confidence in the location and dispersion of the parameter. In using choices (2) and (3) for distr, the distr.opt list must be supplied with mean and sd as subcomponents for those parameters not using the uniform (the examples section hopefully clarifies the usage).
## Not run: # [Example 1] # Distributions of Electrons on a Sphere Problem: # Given n electrons, find the equilibrium state distribution (of minimal Coulomb # potential) of the electrons positioned on a conducting sphere. This model is # from the COPS benchmarking suite. See http://www-unix.mcs.anl.gov/~more/cops/. gofn = function(dat, n) { x = dat[1:n] y = dat[(n+1):(2*n)] z = dat[(2*n+1):(3*n)] ii = matrix(1:n, ncol = n, nrow = n, byrow = TRUE) jj = matrix(1:n, ncol = n, nrow = n) ij = which(ii<jj, arr.ind = TRUE) i = ij[,1] j = ij[,2] # Coulomb potential potential = sum(1.0/sqrt((x[i]-x[j])^2 + (y[i]-y[j])^2 + (z[i]-z[j])^2)) potential } goeqfn = function(dat, n) { x = dat[1:n] y = dat[(n+1):(2*n)] z = dat[(2*n+1):(3*n)] apply(cbind(x^2, y^2, z^2), 1, "sum") } n = 25 LB = rep(-1, 3*n) UB = rep(1, 3*n) eqB = rep(1, n) ans = gosolnp(pars = NULL, fixed = NULL, fun = gofn, eqfun = goeqfn, eqB = eqB, LB = LB, UB = UB, control = list(outer.iter = 100, trace = 1), distr = rep(1, length(LB)), distr.opt = list(), n.restarts = 2, n.sim = 20000, rseed = 443, n = 25) # should get a function value around 243.813 # [Example 2] # Parallel functionality for solving the Upper to Lower CVaR problem (not properly # formulated...for illustration purposes only). mu =c(1.607464e-04, 1.686867e-04, 3.057877e-04, 1.149289e-04, 7.956294e-05) sigma = c(0.02307198,0.02307127,0.01953382,0.02414608,0.02736053) R = matrix(c(1, 0.408, 0.356, 0.347, 0.378, 0.408, 1, 0.385, 0.565, 0.578, 0.356, 0.385, 1, 0.315, 0.332, 0.347, 0.565, 0.315, 1, 0.662, 0.378, 0.578, 0.332, 0.662, 1), 5,5, byrow=TRUE) # Generate Random deviates from the multivariate Student distribution set.seed(1101) v = sqrt(rchisq(10000, 5)/5) S = chol(R) S = matrix(rnorm(10000 * 5), 10000) %*% S ret = S/v RT = as.matrix(t(apply(ret, 1, FUN = function(x) x*sigma+mu))) # setup the functions .VaR = function(x, alpha = 0.05) { VaR = quantile(x, probs = alpha, type = 1) VaR } .CVaR = function(x, alpha = 0.05) { VaR = .VaR(x, alpha) X = as.vector(x[, 1]) CVaR = VaR - 0.5 * mean(((VaR-X) + abs(VaR-X))) / alpha CVaR } .fn1 = function(x,ret) { port=ret%*%x obj=-.CVaR(-port)/.CVaR(port) return(obj) } # abs(sum) of weights ==1 .eqn1 = function(x,ret) { sum(abs(x)) } LB=rep(0,5) UB=rep(1,5) pars=rep(1/5,5) ctrl = list(delta = 1e-10, tol = 1e-8, trace = 0) cl = makePSOCKcluster(2) # export the auxilliary functions which are used and cannot be seen by gosolnp clusterExport(cl, c(".CVaR", ".VaR")) ans = gosolnp(pars, fun = .fn1, eqfun = .eqn1, eqB = 1, LB = LB, UB = UB, n.restarts = 2, n.sim=500, cluster = cl, ret = RT) ans # don't forget to stop the cluster! stopCluster(cl) ## End(Not run)
Useful links