For the problem (P), assume the functions \(f_0,\cdots , f_m\) and \(h_1,\cdots , h_p\) are all differentiable with open domains.

Suppose \(x^∗\) is primal optimal, \((\lambda^∗, v^∗)\) is dual optimal, and there is zero duality gap, then

- \(f_i(x^*) \le 0, i=1,\cdots,m\) and \(h_i(x^*) = 0, i=1,\cdots,p\);
- \(v^* \ge 0\);
- \(\delta f_0(x^*) + \sum_{i=1}^{m} v_i^{*} \delta f_i(x^*) + \sum_{i=1}^{n} \lambda_i^{*} \delta h_i(x^*)=0\);
- \(v_i^{*} f_i(x^*) = 0, i=1,\cdots,m\).

These four conditions, combined together, are called the Karush-Kuhn-Tucker (KKT) conditions.

For the problem (cP), the converse is also true: for (cP), if \(x^∗\) and \((\lambda^*,v^*)\) satisfy the KKT conditions, then \(x^∗\) and \((\lambda^*, v^*)\) are primal and dual optimal, and the duality gap is zero.

Consider (cP), and assume Slater’s condition holds. Then the point \(x^∗\) is primal optimal if and only if there exists some \((\lambda^*, v^*)\) that, together with \(x^∗\), satisfy the KKT conditions. (Only need to think about the \(\to\) direction.)