OptimLib: Sequential Unconstrained Minimization Technique (SUMT)

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Function definition:

bool sumt(arma::vec& init_out_vals, std::function<double (const arma::vec& vals_inp, arma::vec* grad_out, void* opt_data)> opt_objfn, void* opt_data,
          std::function<arma::vec (const arma::vec& vals_inp, arma::mat* jacob_out, void* constr_data)> constr_fn, void* constr_data);

bool sumt(arma::vec& init_out_vals, std::function<double (const arma::vec& vals_inp, arma::vec* grad_out, void* opt_data)> opt_objfn, void* opt_data,
          std::function<arma::vec (const arma::vec& vals_inp, arma::mat* jacob_out, void* constr_data)> constr_fn, void* constr_data, algo_settings_t& settings);

Function arguments:

• init_out_vals a column vector of initial values; will contain the final values.
• opt_objfn the function to be minimized, taking three arguments:
  • vals_inp a vector of inputs;
  • grad_out a vector to store the gradient; and
  • opt_data additional parameters passed to the function.
• opt_data additional parameters passed to the function.
• constr_fn the constraint functions in vector form, taking three arguments:
  • vals_inp a vector of inputs;
  • jacob_out a matrix to store the jacobian; and
  • constr_data additional parameters passed to the constraints.
• constr_data additional parameters passed to the constraints.
• settings parameters controlling the optimization routine; see below.

Optimization control parameters:

• double err_tol the value controlling how small $\| f \|$ should be before 'convergence' is declared.
• int iter_max the maximum number of iterations/updates before the algorithm exits.
• bool vals_bound whether the search space is bounded. If true, then
• arma::vec lower_bounds this defines the lower bounds.
• arma::vec upper_bounds this defines the upper bounds.

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Details:

The SUMT solves the augmented problem

$$\min_x \left\{ f(x) + c \times \dfrac{1}{2} \sum_{k=1}^K \left( \max(0, g_k(x)) \right)^2 \right\} =: \min_x h(x)$$

where $g_k \leq 0$.

The algorithm stops when $\| \nabla h \|$ is less than err_tol, or the total number of 'generations' exceeds a desired (or default) value.