R: Sequential Quadratic Programming (SQP)

slsqp {nloptr} | R Documentation

## Sequential Quadratic Programming (SQP)

### Description

Sequential (least-squares) quadratic programming (SQP) algorithm for nonlinearly constrained, gradient-based optimization, supporting both equality and inequality constraints.

### Usage

    slsqp(x0, fn, gr = NULL, lower = NULL, upper = NULL, hin = NULL,
      hinjac = NULL, heq = NULL, heqjac = NULL, nl.info = FALSE,
      control = list(), ...)

### Arguments

`x0` | starting point for searching the optimum.
`fn` | objective function that is to be minimized.
`gr` | gradient of function `fn`; will be calculated numerically if not specified.
`lower, upper` | lower and upper bound constraints.
`hin` | function defining the inequality constraints, that is `hin>=0` for all components.
`hinjac` | Jacobian of function `hin`; will be calculated numerically if not specified.
`heq` | function defining the equality constraints, that is `heq==0` for all components.
`heqjac` | Jacobian of function `heq`; will be calculated numerically if not specified.
`nl.info` | logical; shall the original NLopt info been shown.
`control` | list of options, see `nl.opts` for help.
`...` | additional arguments passed to the function.

### Details

The algorithm optimizes successive second-order (quadratic/least-squares) approximations of the objective function (via BFGS updates), with first-order (affine) approximations of the constraints.

### Value

List with components:

`par` | the optimal solution found so far.
`value` | the function value corresponding to `par`.
`iter` | number of (outer) iterations, see `maxeval`.
`convergence` | integer code indicating successful completion (> 1) or a possible error number (< 0).
`message` | character string produced by NLopt and giving additional information.

### Note

See more infos at http://ab-initio.mit.edu/wiki/index.php/NLopt_Algorithms.

### Author(s)

Hans W. Borchers

### References

Dieter Kraft, “A software package for sequential quadratic programming”, Technical Report DFVLR-FB 88-28, Institut fuer Dynamik der Flugsysteme, Oberpfaffenhofen, July 1988.

### See Also

`alabama::auglag`, `Rsolnp::solnp`, `Rdonlp2::donlp2`

### Examples

    ##  Solve the Hock-Schittkowski problem no. 100
    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)
    }

    S <- slsqp(x0.hs100, fn = fn.hs100,     # no gradients and jacobians provided
               hin = hin.hs100,
               control = list(xtol_rel = 1e-8, check_derivatives = TRUE))
    S
    ## Optimal value of objective function:  690.622270249131   *** WRONG ***

    # Even the numerical derivatives seem to be too tight.
    # Let's try with a less accurate jacobian.

    hinjac.hs100 <- function(x) nl.jacobian(x, hin.hs100, heps = 1e-2)
    S <- slsqp(x0.hs100, fn = fn.hs100,
               hin = hin.hs100, hinjac = hinjac.hs100,
               control = list(xtol_rel = 1e-8))
    S
    ## Optimal value of objective function:  680.630057392593   *** CORRECT ***

[Package nloptr version 1.2.2.2 Index]