User guide · 2.0 development

Understanding solver results¶

Feasible solutions and bounds¶

A primal solution assigns values to the variables while satisfying the constraints and integrality requirements within tolerances. For minimization, its objective gives an upper bound on the optimum. A valid dual bound is a lower bound. For maximization, the directions reverse.

The gap measures the distance between these bounds. A small gap is useful only together with the feasibility and optimality tolerances and the solver’s reported status. Reaching a time or iteration limit does not establish optimality.

For primal bound p and globally valid dual bound d, the absolute gap is abs(d - p) and the relative gap is abs(d - p) / (1e-10 + abs(p)). Relative gaps are fractions: 0.001 is 0.1%. Either objective-gap tolerance can establish gap convergence. An unavailable bound gives an infinite gap.

getCurrentDualBound() describes the current dual problem. For nonconvex problems it can cease to be globally valid after cuts or repairs. Use getGlobalDualBound() when reporting a valid bound. The public getAbsoluteObjectiveGap() and getRelativeObjectiveGap() use the global bound. The output guide explains the corresponding * marker and why iteration and final gaps can differ.

Model status and termination¶

getModelReturnStatus() describes what was established about the model; getTerminationReason() describes why the algorithm stopped. Inspect both.

Model status	Interpretation
`OptimalGlobal`	The incumbent and global bound meet an objective-gap tolerance. The guarantee depends on any user-specified convexity assumptions.
`FeasibleSolution`	A primal solution exists, but the global gap was not closed. This can follow a limit, stagnation, a user stop, or an error. Inspect the termination reason.
`InfeasibleGlobal`	Infeasibility was established globally.
`InfeasibleLocal`	Infeasibility was reported without a global proof for the nonconvex model.
`NoSolutionReturned`	No feasible solution was returned; this alone is not proof of infeasibility.
`Unbounded`, `UnboundedNoSolution`	Unbounded status, with or without an available primal solution respectively.
`ErrorUnknown`, `ErrorNoSolution`	Error status; inspect solution availability and diagnostics.
`NotSet`	No model status has been established.

Termination reasons include AbsoluteGap, RelativeGap, ConstraintTolerance, TimeLimit, IterationLimit, ObjectiveStagnation, NoDualCutsAdded, InfeasibleProblem, UnboundedProblem, NumericIssues, Error, and UserAbort. Reaching a constraint tolerance or stopping for stagnation is not the same as closing the global objective gap.

A successful solveProblem() call says the solve completed; it does not establish optimality. In Python, a solver failure can raise an exception. Check hasPrimalSolution() before requesting the incumbent. This example is a complete solve with status and bound reporting:

import SHOTpy

solver = SHOTpy.Solver()
solver.updateSetting("Termination.TimeLimit", 60.0)
problem = SHOTpy.Problem(solver)
x = problem.addVariable("x", SHOTpy.VariableType.Integer, 0.0, 5.0)
problem.setObjective((x - 2.3)**2)
problem.finalize()
assert solver.setProblem(problem)
assert solver.solveProblem()

print("Model status:", solver.getModelReturnStatus())
print("Termination reason:", solver.getTerminationReason())
print("Global dual bound:", solver.getGlobalDualBound())
if solver.hasPrimalSolution():
    solution = solver.getPrimalSolution()
    print("Feasible objective:", solution.objValue)
    print("Original variable values:", solution.point)
    print("Global relative gap:", solver.getRelativeObjectiveGap())
else:
    print("No feasible solution returned")

C++ exposes the same methods and uses E_ModelReturnStatus and E_TerminationReason; see the integration example. In file-based workflows, inspect the OSrL/modeling-system status. The executable’s exit code 0 only says that the run completed. Pyomo’s mapping of AMPL statuses also requires care; see command-line usage.

Outer approximation¶

SHOT constructs linear approximations of nonlinear expressions and solves mixed-integer subproblems. Primal searches seek feasible solutions to the original nonlinear problem. Convexity determines when linearizations provide globally valid bounds; nonconvex models require additional care and do not generally have the same guarantees. See modeling and reformulation for domain restrictions, bounds, and how the original model is transformed.

Convexity specified by the user¶

SHOT detects the convexity of the objective function and of each constraint function, and regards a function as nonconvex when it cannot prove otherwise. When you know better, you can specify the convexity with setFunctionConvexity() on a quadratic or nonlinear constraint or objective function, in C++ and SHOTpy, or with the functionConvexity argument of Problem.addConstraint() and Problem.setObjective() in SHOTpy. The value is Convex, Concave or Nonconvex. For a constraint L <= f(x) <= U, it is the convexity of f(x), and the convexity of the constraint follows from it and the bounds: f(x) <= U is convex for a convex function, and L <= f(x) for a concave one. A minimized convex or maximized concave objective function gives a convex problem. Such a function is not reformulated, e.g., its sums are not partitioned, since the convexity of its parts is not known.

Model.Convexity.AssumeConvex (--convex) instead assumes the whole problem to be convex. In both cases, the global optimality of a solution depends on the assumption, which the solution report, the UserSpecifiedConvexity entry of the OSrL results and getUserSpecifiedConvexityDescription() tell.

Solver backends¶

The available MIP/NLP backends depend on compilation options and installed licenses. Select a backend available in your build and consult the settings reference. See the compilation guide for dependencies and the debugging guide for intermediate problems and diagnostic logs.