Whether the original problem is a convex optimization problem or not, it can be transformed into a convex optimization problem to solve.
When the strong duality of Lagrange dual problem holds, the original problem can be solved by solving the dual problem. When the original problem is a convex optimization problem, strong duality often holds. Otherwise, the lower bound of the optimal value of the original problem can be obtained by solving the dual problem
Many research results have given strong duality conditions other than convexity, which are called constraint criteria.
When the original problem is a convex optimization problem, the point satisfying KKT condition is also the optimal solution of the original problem and dual problem.
If a convex optimization problem has differentiable objective function and constraint function, and satisfies Slater condition, then KKT condition is the necessary and sufficient condition of optimality.
KKT condition plays an important role in the field of optimization. In some cases, the optimization problem can be solved by solving KKT conditions analytically. Lagrange multiplier method in higher algebra can be understood as using KKT condition to solve the constrained extreme value problem. [2]
More generally, many methods for solving convex optimization problems can be understood as methods for solving KKT conditions.
[1], convex optimization, equivalent translation, Wang, equivalent translation.
[2], "Advanced Mathematics" (Tongji Fourth Edition)
Convex Optimization (I) —— A Survey
Convex Optimization (Ⅱ) —— Convex Set
Convex Optimization (Ⅲ) —— Convex Function
Convex Optimization (IV)-Problem Solving
Convex Optimization (V) —— Backtracking straight line search
Convex optimization (ⅵ)-steepest descent method
Convex Optimization (VII) —— Newton Method
Convex Optimization (8)—— Lagrange Dual Problem
20 16-03-0 1 first released
20 16-08-08 modify the title of the article, rearrange and improve it.