SHOT (Supporting Hyperplane Optimization Toolkit) is a deterministic solver for mixed-integer nonlinear programming problems (MINLPs).

Originally, SHOT was intended for convex MINLP problems only, but now 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.

SHOT has mainly been developed by Andreas Lundell (Åbo Akademi University, Finland) and Jan Kronqvist (Imperial College London, UK). For more details, see [167, 163, 148, 149].

SHOT supports GAMS equations that use the following intrinsic functions: abs, cos, cvPower, div, exp, log, log10, log2, pi, power, rPower, sin, sqr, sqrt, vcPower.

Algorithm

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 a mixed-integer linear programming (MIP) solver. GAMS/SHOT uses CPLEX, if a GAMS/CPLEX license is available, and otherwise CBC. Users with a license from Gurobi can also select Gurobi as MIP solver. If CPLEX or Gurobi is used, the subproblems can also include quadratic and bilinear nonlinearities directly.

The solution to the outer approximation problem provides a dual bound (i.e., a lower bound when solving a minimization problem) to the optimal value of the original problem if it 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. [163]).

To get a primal bound (i.e., an upper bound when solving a minimization problem) on the optimal value, 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 NLP solver, which is either Ipopt or one of the GAMS NLP solvers.
• By checking solutions from the MIP solver's solution pool for points that fulfill also the nonlinear constraints in the original MINLP problem.
• By performing root searches.

When a termination criterion like a tolerance on the relative or absolute objective gap or a time limit is fulfilled, SHOT terminates and returns the current primal solution to GAMS. If the original problem is convex and SHOT could close the objective gap, then this is a global optimal solution to the problem. If it is nonconvex, then there is normally no guarantee that such a solution can be found. However, SHOT will always, in addition to a primal solution, return a valid dual bound on the solution in model attribute objest, unless Model.Convexity.AssumeConvex has been enabled.

Usage

The following statement can be used inside your GAMS program to specify using SHOT

Option MINLP = SHOT;     { or MIQCP }

The above statement should appear before the Solve statement. If SHOT was specified as the default MINLP or MIQCP solver during GAMS installation, the above statement is not necessary.

Specification of SHOT Options

GAMS/SHOT supports the GAMS parameters reslim, iterlim, nodlim, optcr, optca, cutoff, and threads.

Options can be specified by a SHOT options file. A SHOT options file consists of one option or comment per line. An asterik (*) at the beginning of a line causes the entire line to be ignored. Otherwise, the line will be interpreted as an option name and value separated by an equal sign (=) and any amount of white space (blanks or tabs).

A small example for a shot.opt file is:

Dual.CutStrategy = 1
Dual.MIP.Solver = 2
Output.Console.DualSolver.Show = true

It causes GAMS/SHOT to use the Extended Cutting Plane (ECP) method instead of the Extended Supporting Hyperplane (EHP) method, changes the MIP solver to CBC, and enables showing the output of the solver that computes dual bounds (typically the MIP solver).

Attention

SHOT requires options to be specified using exactly the names as specified in the documentation. That is, also casing matters.

List of SHOT Options

In the following, we give a detailed list of all SHOT options.

Dual strategy

These settings control the various functionality of the dual strategy in SHOT, i.e., the polyhedral outer approximation utilizing the ESH or ECP algorithms.

Optimization model

These settings control various aspects of SHOT's representation for and handling of the provided optimization model.

Solver output

These settings control how much and what output is shown to the user from the solver.

Primal heuristics

These settings control the primal heuristics used in SHOT.

Strategy

Overall strategy parameters used in SHOT.

Subsolver functionality

These settings allow for more direct control of the different subsolvers utilized in SHOT.

Termination

These settings control when SHOT will terminate the solution process.