gemseo_mma / opt / core

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mma module

GCMMA-MMA-Python. This file is part of GCMMA-MMA-Python.

GCMMA-MMA-Python is licensed under the terms of GNU General Public License as published by the Free Software Foundation. For more information and the LICENSE file, see [here](https://github.com/arjendeetman/GCMMA-MMA-Python). The original work is written by Krister Svanberg in MATLAB. This is the python version of the code written Arjen Deetman. version 09-11-2019.

MMA optimizer.

Original work written by Krister Svanberg in Matlab. This is the python version of the code written by Arjen Deetman.

gemseo_mma.opt.core.mma.compute_kkt_residual_on_local_approximation(m, n, x, y, z, lam, xsi, eta, mu, zet, s, xmin, xmax, df0dx, fval, dfdx, a0, a, c, d)[source]

KKT residual computation.

The left hand sides of the KKT conditions for the following nonlinear programming problem are calculated.

Minimize f_0(x) + a_0*z + sum(c_i*y_i + 0.5*d_i*(y_i)^2) subject to f_i(x) - a_i*z - y_i <= 0, i = 1,…,m

xmax_j <= x_j <= xmin_j, j = 1,…,n z >= 0, y_i >= 0, i = 1,…,m

Parameters:

  • m (int) – The number of general constraints.

  • n (int) – The number of variables x_j.

  • x (ndarray) – The current values of the n variables x_j.

  • y (ndarray) – The current values of the m variables y_i.

  • z (ndarray) – The current value of the single variable z.

  • lam (ndarray) – The Lagrange multipliers for the m general constraints.

  • xsi (ndarray) – The Lagrange multipliers for the n constraints xmin_j - x_j <= 0.

  • eta (ndarray) – The Lagrange multipliers for the n constraints x_j - xmax_j <= 0.

  • mu (ndarray) – The Lagrange multipliers for the m constraints -y_i <= 0.

  • zet (ndarray) – The Lagrange multiplier for the single constraint -z <= 0.

  • s (ndarray) – The Slack variables for the m general constraints.

  • xmin (ndarray) – The Lower bounds for the variables x_j.

  • xmax (ndarray) – The Upper bounds for the variables x_j.

  • df0dx (ndarray) – The vector with the derivatives of the objective function f_0 with respect to the variables x_j, calculated at x.

  • fval (ndarray) – The vector with the values of the constraint functions f_i, calculated

  • x. (at) –

  • dfdx (ndarray) – The (m x n)-matrix with the derivatives of the constraint functions f_i with respect to the variables x_j, calculated at x. dfdx(i,j) = the derivative of f_i with respect to x_j.

  • a0 (float) – The constants a_0 in the term a_0*z.

  • a (ndarray) – The vector with the constants a_i in the terms a_i*z.

  • c (ndarray) – The vector with the constants c_i in the terms c_i*y_i.

  • d (ndarray) – The vector with the constants d_i in the terms 0.5*d_i*(y_i)^2.

Returns:

The vector residual, its norm and its maximum.

Return type:

tuple[ndarray, ndarray, ndarray]

gemseo_mma.opt.core.mma.solve_mma_local_approximation_problem(m, n, n_iterations, xval, xmin, xmax, xold1, xold2, f0val, df0dx, fval, dfdx, low, upp, a0, a, c, d, move, external_move_limit=10.0, internal_limit=0.01, asyinit=0.5, asyincr=1.2, asydecr=0.7)[source]

MMA sub function.

This function mmasub performs one MMA-iteration, aimed at solving the nonlinear programming problem:

Minimize f_0(x) + a_0*z + sum( c_i*y_i + 0.5*d_i*(y_i)^2 ) subject to f_i(x) - a_i*z - y_i <= 0, i = 1,…,m

xmin_j <= x_j <= xmax_j, j = 1,…,n z >= 0, y_i >= 0, i = 1,…,m

Parameters:

  • m (int) – The number of general constraints.

  • n (int) – The number of variables x_j.

  • n_iterations (int) – The current iteration number (=1 the first time mmasub is called).

  • xval (ndarray) – The column vector with the current values of the variables x_j.

  • xmin (ndarray) – The column vector with the lower bounds for the variables x_j.

  • xmax (ndarray) – The column vector with the upper bounds for the variables x_j.

  • xold1 (ndarray) – The value of xval, one iteration ago (provided that n_iterations>1).

  • xold2 (ndarray) – The value of xval, two iterations ago (provided that n_iterations>2).

  • f0val (ndarray) – The value of the objective function f_0 at xval.

  • df0dx (ndarray) – The column vector with the derivatives of the objective function f_0 with respect to the variables x_j, calculated at xval.

  • fval (ndarray) – The column vector with the values of the constraint functions f_i, calculated at xval.

  • dfdx (ndarray) – The (m x n)-matrix with the derivatives of the constraint functions f_i with respect to the variables x_j, calculated at xval.

  • low (ndarray) – The column vector with the lower asymptotes from the previous iteration (provided that n_iterations>1).

  • upp (ndarray) – The column vector with the upper asymptotes from the previous iteration (provided that n_iterations>1).

  • a0 (float) – The constants a_0 in the term a_0*z.

  • a (ndarray) – The column vector with the constants a_i in the terms a_i*z.

  • c (ndarray) – The column vector with the constants c_i in the terms c_i*y_i.

  • d (ndarray) – The column vector with the constants d_i in the terms 0.5*d_i*(y_i)^2.

  • move (float) – The maximum optimization step.

  • external_move_limit (float) –

    The maximum distance of the asymptotes from the current design variable value.

    By default it is set to 10.0.

  • internal_limit (float) –

    The minimum distance of the asymptotes from the current design variable value.

    By default it is set to 0.01.

  • asyinit (float) –

    The initial asymptotes distance from the current design variable value.

    By default it is set to 0.5.

  • asyincr (float) –

    The incremental factor for successful iterations.

    By default it is set to 1.2.

  • asydecr (float) –

    The decremental factor for unsuccessful iterations.

    By default it is set to 0.7.

Returns:

The Column vector with the optimal values of the variables x_j in the current MMA subproblem.

The Column vector with the optimal values of the variables y_i in the current MMA subproblem.

The Scalar with the optimal value of the variable z in the current MMA subproblem.

The Lagrange multipliers for the m general MMA constraints.

The Lagrange multipliers for the n constraints alfa_j - x_j <= 0.

The Lagrange multipliers for the n constraints x_j - beta_j <= 0.

The Lagrange multipliers for the m constraints -y_i <= 0.

The Lagrange multiplier for the single constraint -z <= 0.

The Slack variables for the m general MMA constraints.

The Column vector with the lower asymptotes, calculated and used in the current MMA subproblem.

The Column vector with the upper asymptotes, calculated and used in the current MMA subproblem.

Return type:

tuple[ndarray, ndarray, ndarray, ndarray, ndarray, ndarray, ndarray, ndarray, ndarray, ndarray, ndarray]