The system of equations and inequalities corresponding to the KKT conditions is usually not solved directly, except in the few special cases where a closed-form solution can be derived analytically. In general, many optimization algorithms can be interpreted as methods for numerically solving the KKT system of equations and inequalities. Suppose that the objective function formula_30 and the constraint functions formula_31 and formula_32 are continuously differentiable at a point formula_33. If formula_34 is a local optimum and the optimization problem satisfies some regularity conditions (see below), then there exist constants formula_35 and formula_36, called KKT multipliers, such that the following four groups of conditions hold:      In the particular case formula_45, i.e., when there are no inequality constraints, the KKT conditions turn into the Lagrange conditions, and the KKT multipliers are called Lagrange multipliers. If some of the functions are non-differentiable, subdifferential versions of Karush–Kuhn–Tucker (KKT) conditions are available. The necessary conditions can be written with Jacobian matrices of the constraint functions. Let formula_46 be defined as formula_47 and let formula_48 be defined as formula_49. Let formula_50 and formula_51. Then the necessary conditions can be written as:      In order for a minimum point formula_34 to satisfy the above KKT conditions, the problem should satisfy some regularity conditions; some common examples are tabulated here:  It can be shown that  and  (and the converses are not true), although MFCQ is not equivalent to CRCQ. In practice weaker constraint qualifications are preferred since they provide stronger optimality conditions. In some cases, the necessary conditions are also sufficient for optimality. In general, the necessary conditions are not sufficient for optimality and additional information is required, such as the Second Order Sufficient Conditions (SOSC). For smooth functions, SOSC involve the second derivatives, which explains its name. The necessary conditions are sufficient for optimality if the objective function formula_6 of a maximization problem is a concave function, the inequality constraints formula_62 are continuously differentiable convex functions and the equality constraints formula_63 are affine functions. Similarly, if the objective function formula_6 of a minimization problem is a convex function, the necessary conditions are also sufficient for optimality. It was shown by Martin in 1985 that the broader class of functions in which KKT conditions guarantees global optimality are the so-called Type 1 invex functions.