QuickStart

Snopt solves optimization problems of the following form:

\[\begin{aligned} \text{minimize} &\quad f(x) \\ \text{subject to} &\quad l_g \le g(x) \le u_g \\ &\quad l_x \le x \le u_x \end{aligned}\]

Equality constraints can be specified by setting ${l_g}_i = {u_g}_i$. The expected function signature is:

f = func!(g, x)

where $g$ is modified in-place. The optimizer returns $x^*, f^*$, an information message regarding how the algorithm terminated, and potentially a struct with additional solver-specific outputs.

We start by loading the package.

using SNOW

Next, we define the function we wish to optimize.

function simple!(g, x)
    # objective
    f = 4*x[1]^2 - x[1] - x[2] - 2.5

    # constraints
    g[1] = -x[2]^2 + 1.5*x[1]^2 - 2*x[1] + 1
    g[2] = x[2]^2 + 2*x[1]^2 - 2*x[1] - 4.25
    g[3] = x[1]^2 + x[1]*x[2] - 10.0

    return f
end

We can now optimize the function as follows:

x0 = [1.0; 2.0]  # starting point
ng = 3  # number of constraints
xopt, fopt, info = minimize(simple!, x0, ng)

While the defaults are suitable in this case, for the purposes of demonstration we will be more explicit about the bounds on x (defaults to -Inf to +Inf) and the bounds on the nonlinear constraints (defaults to g(x) <= 0). We will also explicitly set the IPOPT optimizer (also a default).

x0 = [1.0; 2.0]  # starting point
lx = [-5.0, -5]  # lower bounds on x
ux = [5.0, 5]  # upper bounds on x
ng = 3  # number of constraints
lg = -Inf*ones(ng)  # lower bounds on g
ug = zeros(ng)  # upper bounds on g
options = Options(solver=IPOPT())  # choosing IPOPT solver

xopt, fopt, info = minimize(simple!, x0, ng, lx, ux, lg, ug, options)

println("xstar = ", xopt)
println("fstar = ", fopt)
println("info = ", info)

This is Ipopt version 3.13.2, running with linear solver mumps.
NOTE: Other linear solvers might be more efficient (see Ipopt documentation).

Number of nonzeros in equality constraint Jacobian...:        0
Number of nonzeros in inequality constraint Jacobian.:        6
Number of nonzeros in Lagrangian Hessian.............:        0

Total number of variables............................:        2
                     variables with only lower bounds:        0
                variables with lower and upper bounds:        2
                     variables with only upper bounds:        0
Total number of equality constraints.................:        0
Total number of inequality constraints...............:        3
        inequality constraints with only lower bounds:        0
   inequality constraints with lower and upper bounds:        0
        inequality constraints with only upper bounds:        3

iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
   0 -1.5000000e+00 0.00e+00 2.62e+00   0.0 0.00e+00    -  0.00e+00 0.00e+00   0
   1 -2.7235302e+00 4.21e+00 1.74e+00   0.0 6.05e+00    -  5.62e-01 1.00e+00f  1
   2 -9.4251777e-01 1.99e+00 6.34e+00  -0.6 8.92e+00    -  5.92e-01 5.44e-01H  1
   3 -4.2494716e+00 2.33e+00 4.45e+00  -0.3 7.06e+00    -  1.00e+00 1.00e+00f  1
   4 -4.8239643e+00 6.84e-01 8.74e-01  -0.5 3.10e+00    -  1.00e+00 9.63e-01h  1
   5 -4.6648335e+00 0.00e+00 1.04e+00  -1.8 1.21e+00    -  9.80e-01 1.00e+00h  1
   6 -4.6800544e+00 0.00e+00 6.43e-02  -3.1 1.41e-01    -  1.00e+00 1.00e+00h  1
   7 -4.6833853e+00 0.00e+00 1.42e-03  -4.7 1.87e-02    -  1.00e+00 1.00e+00h  1
   8 -4.6834353e+00 0.00e+00 2.57e-06  -6.6 3.32e-04    -  1.00e+00 1.00e+00h  1
   9 -4.6834354e+00 0.00e+00 3.14e-07 -11.0 2.79e-06    -  1.00e+00 1.00e+00h  1
iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
  10 -4.6834354e+00 0.00e+00 1.34e-08 -11.0 4.90e-07    -  1.00e+00 1.00e+00h  1
  11 -4.6834354e+00 0.00e+00 4.71e-09 -11.0 1.50e-08    -  1.00e+00 1.00e+00h  1

Number of Iterations....: 11

                                   (scaled)                 (unscaled)
Objective...............:  -4.6834354345613223e+00   -4.6834354345613223e+00
Dual infeasibility......:   4.7113362378059520e-09    4.7113362378059520e-09
Constraint violation....:   0.0000000000000000e+00    0.0000000000000000e+00
Complementarity.........:   1.0000000000000041e-11    1.0000000000000041e-11
Overall NLP error.......:   4.7113362378059520e-09    4.7113362378059520e-09


Number of objective function evaluations             = 13
Number of objective gradient evaluations             = 12
Number of equality constraint evaluations            = 0
Number of inequality constraint evaluations          = 13
Number of equality constraint Jacobian evaluations   = 0
Number of inequality constraint Jacobian evaluations = 12
Number of Lagrangian Hessian evaluations             = 0
Total CPU secs in IPOPT (w/o function evaluations)   =      0.127
Total CPU secs in NLP function evaluations           =      0.003

EXIT: Optimal Solution Found.
xstar = [0.1644378582645418, 2.127156813219299]
fstar = -4.683435434561322
info = Solve_Succeeded