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In this paper we consider three minimization problems, namely quadratic,$\rho$-convex and quadratic fractional programing problems. The quadraticproblem is considered with quadratic inequality constraints with boundedcontinuous and discrete mixed variables. The $\rho$-convex problem isconsidered with $\rho$-convex inequality constraints in mixed variables. Thequadratic fractional problem is studied with quadratic fractional constraintsin mixed variables. For all three problems we reformulate the problem as amathematical programming problem and apply standard Karush Kuhn Tuckernecessary conditions. Then, for each problem, we provide local necessaryoptimality condition. Further, for each problem a Lagrangian multipliersufficient optimality condition is provided to identify global minimizer amongthe local minimizers. For the quadratic problem underestimation of a Lagrangianwas employed to obtain the desired sufficient conditions. For the $\rho$-convexproblem we obtain two sufficient optimality conditions to distinguish a globalminimizer among the local minimizers, one with an underestimation of aLagrangian and the other with a different technique. A global sufficientoptimality condition for the quadratic fractional problem is obtained byreformulating the problem as a quadratic problem and then utilizing the resultsof the quadratic problem. Examples are provided to illustrate the significanceof the results obtained.

Lagrange Multiplier Local Necessary and Global Sufficiency Criteria for Some Non-Convex Programming Problems