This means that the Solver has found the optimal or “best” solution under the circumstances.  The exact meaning depends on whether you are solving a linear or quadratic, smooth nonlinear, global optimization, or integer programming problem, as outlined below.  Solvers for non-smooth problems rarely if ever display this message, because they have no way of testing the solution for true optimality.

If you are solving a linear programming problem or a convex quadratic programming problem with the LP/Quadratic Solver, the Solver has found the globally optimal solution:  There is no other solution satisfying the constraints that has a better value for the objective.  It is possible that there are other solutions with the same objective value, but all such solutions are linear combinations of the current decision variable values.

If you are solving a linear (LP), convex quadratic (QP) or quadratically constrained (QCP), or second order cone programming (SOCP) problem with the SOCP Barrier Solver, the Solver has found the globally optimal solution:  There is no other solution satisfying the constraints that has a better value for the objective.  It’s possible that there are other solutions with the same objective value, but all such solutions are linear combinations of the current decision variable values.

If you are solving a smooth nonlinear optimization problem with no integer constraints, the GRG Solver has found a locally optimal solution:  There is no other set of values for the decision variables close to the current values and satisfying the constraints that yields a better value for the objective.  In general, there may be other sets of values for the variables, far away from the current values, which yield better values for the objective and still satisfy the constraints.

If you are using the Interval Global Solver (within the Analytic Solver products) for global optimization of a smooth nonlinear problem with no integer constraints, this means that the Solver has found the globally optimal solution:  There is no other solution satisfying the constraints that has a better value for the objective.  But this is subject to limitations due to the finite precision of computer arithmetic – discussed below in “Limitations on Global Optimization” – that can, in rare cases, cause the Solver to “miss” a feasible solution with an even better objective value.

If you are solving a mixed-integer programming problem (any problem with integer constraints), this message means that the Branch & Bound method has found a solution satisfying the constraints (including the integer constraints) with the “best possible” objective value (but see the next paragraph).  If the problem is linear or quadratic, the true integer optimal solution has been found.  If the problem is smooth nonlinear, the Branch & Bound process has found the best of the locally optimal solutions found for subproblems by the nonlinear Solver.

In the standard Microsoft Excel Solver, this message also appears for mixed-integer problems where the Solver stopped because the solution was within the range of the true integer optimal solution allowed by the Tolerance value in the Solver Options dialog (5% by default).  When the Branch & Bound process stops due to a nonzero Tolerance without “proving optimality,” the message “Solver found an integer solution within tolerance.  All constraints are satisfied” (result code 14) is displayed to distinguish this condition (see below).