Algorithm

SHOT is a software for solving mathematical optimization problems of the mixed-integer nonlinear programming (MINLP) class.

Originally SHOT was intended for convex MINLP problems only, but as of version 1.0 it also has functionality to solve nonconvex MINLP problems, as a heuristic method without providing any guarantees of global optimality. SHOT can solve certain nonconvex problem types to global optimality as well.

In contrast to other polyhedral outer approximation solvers, the bounds on the objective function value provided by SHOT are valid also for nonconvex problems (with the exception when using the

`Model.Convexity.AssumeConvex=true`

setting for nonconvex problems).SHOT is based on iteratively creating a tighter polyhedral approximation of the nonlinear feasible set by generating supporting hyperplanes or cutting planes. These linearized problems are then solved with an mixed-integer linear programming (MILP) solver such as CPLEX, Gurobi or Cbc. If CPLEX or Gurobi is used, the subproblems can also include quadratic and bilinear nonlinearities directly; then MIQP or MIQCQP subproblems are solved.

The solution to the outer approximation problem provides a lower (dual) bound (when solving a minimization problem) to the original problem if the problem is convex. If the problem is nonconvex, convergence to the global optimal solution cannot be guaranteed (but might be achieved for certain classes of problems, cf. http://www.optimization-online.org/DB_HTML/2020/03/7691.html).

To get an upper (primal) bound (when solving a minimization problem) on the optimal solution SHOT utilizes the following heuristics:

- Solving nonlinear programming (NLP) problems where the integer variables have been fixed to valid values. This is done by calling an external NLP solver (e.g. Ipopt).
- By checking solutions from the MIP solver's solution pool for points that fulfill also the nonlinearities in the original MINLP problem.
- By performing root searches.

When the relative or absolute difference (objective gap) between the primal and dual bounds is less than a user-specified value, SHOT terminates with the current primal solution. If the original problem is convex, this is a global solution to the problem. If it is nonconvex, there is normally no guarantee that such a solution can be found, however SHOT will always in addition to the primal solution give a valid lower bound on the solution.

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Dual bound through polyhedral (outer) approximation

Primal bound using heuristics

Termination