Conservative Convex Separable Approximation with Affine Approximation plus Quadratic Penalty
This is a variant of CCSA ("conservative convex separable approximation") which, instead of constructing local MMA approximations, constructs simple quadratic approximations (or rather, affine approximations plus a quadratic penalty term to stay conservative)
ccsaq(x0, fn, gr = NULL, lower = NULL, upper = NULL, hin = NULL, hinjac = NULL, nl.info = FALSE, control = list(), ...)
x0 |
starting point for searching the optimum. |
fn |
objective function that is to be minimized. |
gr |
gradient of function |
lower, upper |
lower and upper bound constraints. |
hin |
function defining the inequality constraints, that is
|
hinjac |
Jacobian of function |
nl.info |
logical; shall the original NLopt info been shown. |
control |
list of options, see |
... |
additional arguments passed to the function. |
List with components:
par |
the optimal solution found so far. |
value |
the function value corresponding to |
iter |
number of (outer) iterations, see |
convergence |
integer code indicating successful completion (> 1) or a possible error number (< 0). |
message |
character string produced by NLopt and giving additional information. |
“Globally convergent” does not mean that this algorithm converges to the global optimum; it means that it is guaranteed to converge to some local minimum from any feasible starting point.
Krister Svanberg, “A class of globally convergent optimization methods based on conservative convex separable approximations,” SIAM J. Optim. 12 (2), p. 555-573 (2002).
## Solve the Hock-Schittkowski problem no. 100 with analytic gradients x0.hs100 <- c(1, 2, 0, 4, 0, 1, 1) fn.hs100 <- function(x) { (x[1]-10)^2 + 5*(x[2]-12)^2 + x[3]^4 + 3*(x[4]-11)^2 + 10*x[5]^6 + 7*x[6]^2 + x[7]^4 - 4*x[6]*x[7] - 10*x[6] - 8*x[7] } hin.hs100 <- function(x) { h <- numeric(4) h[1] <- 127 - 2*x[1]^2 - 3*x[2]^4 - x[3] - 4*x[4]^2 - 5*x[5] h[2] <- 282 - 7*x[1] - 3*x[2] - 10*x[3]^2 - x[4] + x[5] h[3] <- 196 - 23*x[1] - x[2]^2 - 6*x[6]^2 + 8*x[7] h[4] <- -4*x[1]^2 - x[2]^2 + 3*x[1]*x[2] -2*x[3]^2 - 5*x[6] +11*x[7] return(h) } gr.hs100 <- function(x) { c( 2 * x[1] - 20, 10 * x[2] - 120, 4 * x[3]^3, 6 * x[4] - 66, 60 * x[5]^5, 14 * x[6] - 4 * x[7] - 10, 4 * x[7]^3 - 4 * x[6] - 8 )} hinjac.hs100 <- function(x) { matrix(c(4*x[1], 12*x[2]^3, 1, 8*x[4], 5, 0, 0, 7, 3, 20*x[3], 1, -1, 0, 0, 23, 2*x[2], 0, 0, 0, 12*x[6], -8, 8*x[1]-3*x[2], 2*x[2]-3*x[1], 4*x[3], 0, 0, 5, -11), 4, 7, byrow=TRUE) } # incorrect result with exact jacobian S <- ccsaq(x0.hs100, fn.hs100, gr = gr.hs100, hin = hin.hs100, hinjac = hinjac.hs100, nl.info = TRUE, control = list(xtol_rel = 1e-8)) S <- ccsaq(x0.hs100, fn.hs100, hin = hin.hs100, nl.info = TRUE, control = list(xtol_rel = 1e-8))
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