  In this paper we consider three minimization problems, namely quadratic,
$\rho$-convex and quadratic fractional programing problems. The quadratic
problem is considered with quadratic inequality constraints with bounded
continuous and discrete mixed variables. The $\rho$-convex problem is
considered with $\rho$-convex inequality constraints in mixed variables. The
quadratic fractional problem is studied with quadratic fractional constraints
in mixed variables. For all three problems we reformulate the problem as a
mathematical programming problem and apply standard Karush Kuhn Tucker
necessary conditions. Then, for each problem, we provide local necessary
optimality condition. Further, for each problem a Lagrangian multiplier
sufficient optimality condition is provided to identify global minimizer among
the local minimizers. For the quadratic problem underestimation of a Lagrangian
was employed to obtain the desired sufficient conditions. For the $\rho$-convex
problem we obtain two sufficient optimality conditions to distinguish a global
minimizer among the local minimizers, one with an underestimation of a
Lagrangian and the other with a different technique. A global sufficient
optimality condition for the quadratic fractional problem is obtained by
reformulating the problem as a quadratic problem and then utilizing the results
of the quadratic problem. Examples are provided to illustrate the significance
of the results obtained.
