Karush-Kuhn-Tucker (KKT) Conditions
Main.KuhnTucker History
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Part 1: 5 Minute Tutorial on the KKT Conditions
Part 1: Tutorial on the KKT Conditions
Part 2: 5 Minute Tutorial with KKT Conditions and Inequality Constraints
Part 2: KKT Conditions with Inequality Constraints
Part 3: 5 Minute KKT Exercise with both Inequality and Equality Constraints
Part 3: KKT Exercise with both Inequality and Equality Constraints
Part 4: 5 Minute Application Exercise for the Optimal Volume of a Tank
Part 4: Application Exercise for the Optimal Volume of a Tank
The necessary conditions for a constrained local optimum are called the Karush Kuhn Tucker (KKT) Conditions, and these conditions play a very important role in constrained optimization theory and algorithm development. For an optimization problem:
The optimality conditions for a constrained local optimum are called the Karush Kuhn Tucker (KKT) conditions and they play an important role in constrained optimization theory and algorithm development. The KKT conditions for optimality are a set of necessary conditions for a solution to be optimal in a mathematical optimization problem. They are necessary and sufficient conditions for a local minimum in nonlinear programming problems. The KKT conditions consist of the following elements:
For an optimization problem:
1. Feasible Constraints
1. Primal Feasibility: All constraints must be satisfied.
2. No Feasible Descent
2. No Feasible Descent: No possible objective improvement at the solution.
3. Complementary Slackness
3. Complementarity: The product of the Lagrange multipliers and the corresponding variables must be zero.
4. Positive Lagrange Multipliers
4. Dual Feasibility: The Lagrange multipliers associated with constraints have to be non-negative (zero or positive).
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The feasibility condition (1) applies to both equality and inequality constraints and is simply a statement that the constraints must not be violated at optimal conditions. The gradient condition (2) ensures that there is no feasible direction that could potentially improve the objective function. The last two condition (3 and 4) are only required with inequality constraints and enforce a positive Lagrange multiplier when the constraint is active (=0) and a zero Lagrange multiplier when the constraint is inactive (>0).
The feasibility condition (1) applies to both equality and inequality constraints and is simply a statement that the constraints must not be violated at optimal conditions. The gradient condition (2) ensures that there is no feasible direction that could potentially improve the objective function. The last two conditions (3 and 4) are only required with inequality constraints and enforce a positive Lagrange multiplier when the constraint is active (=0) and a zero Lagrange multiplier when the constraint is inactive (>0).


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There are four KKT conditions for optimality with the primal `x*` variable and dual `\lambda*` variable optimal values.
There are four KKT conditions for optimal primal `(x)` and dual `(\lambda)` variables. The asterisk (*) denotes optimal values.
$$\quad\quad\quad g_i(x)-b_i = 0 \quad i=k+1,\ldots,m$$
$$\quad\quad\quad\quad\quad g_i(x)-b_i = 0 \quad i=k+1,\ldots,m$$
$$g_i(x^*)-b_i$$
$$g_i(x^*)-b_i \mathrm{\;is\;feasible}$$
$$\mathrm{subject\;to}\;g_i(x)-b_i \ge 0 \quad i=1,\ldots,k$$ $$\quad\quad g_i(x)-b_i = 0 \quad i=k+1,\ldots,m$$
$$\mathrm{subject\;to}\quad g_i(x)-b_i \ge 0 \quad i=1,\ldots,k$$ $$\quad\quad\quad g_i(x)-b_i = 0 \quad i=k+1,\ldots,m$$
$$`\lambda_i^* \left( g_i(x^*)-b_i \right) = 0$$
$$\lambda_i^* \left( g_i(x^*)-b_i \right) = 0$$
$$\gradient f(x^*)-\sum_{i=1}^m \lambda_i^* \grad g_i\left(x^*\right)=0$$
$$\nabla f(x^*)-\sum_{i=1}^m \lambda_i^* \nabla g_i\left(x^*\right)=0$$
$$g_i(x*)-b_i$$
$$g_i(x^*)-b_i$$
$$\grad f(x*)-\sum_{i=1}^m \lambda_i^* \grad g_i\left(x*\right)=0$$
$$\gradient f(x^*)-\sum_{i=1}^m \lambda_i^* \grad g_i\left(x^*\right)=0$$
$$`\lambda_i* \left( g_i(x*)-b_i \right) = 0$$
$$`\lambda_i^* \left( g_i(x^*)-b_i \right) = 0$$
$$\lambda_i* \ge 0$$
$$\lambda_i^* \ge 0$$
- `g_i(x*)-b_i` (Feasible constraints)
- `\grad f(x*)-\sum_{i=1}^m \lambda_i^* \grad g_i\left(x*\right)=0` (No direction improves objective and is feasible)
- `\lambda_i* \left( g_i(x*)-b_i \right) = 0` (Complementary slackness)
- `\lambda_i* \ge 0` (Positive Lagrange multipliers)
1. Feasible Constraints
$$g_i(x*)-b_i$$
2. No Feasible Descent
$$\grad f(x*)-\sum_{i=1}^m \lambda_i^* \grad g_i\left(x*\right)=0$$
3. Complementary Slackness
$$`\lambda_i* \left( g_i(x*)-b_i \right) = 0$$
4. Positive Lagrange Multipliers
$$\lambda_i* \ge 0$$
The four KKT conditions for optimality are
- `g_i(x)-b_i` is feasible
- `\grad f(x^*)-\sum_{i=1}^m \lambda_i^* \grad g_i\left(x^*\right)=0`
There are four KKT conditions for optimality with the primal `x*` variable and dual `\lambda*` variable optimal values.
- `g_i(x*)-b_i` (Feasible constraints)
- `\grad f(x*)-\sum_{i=1}^m \lambda_i^* \grad g_i\left(x*\right)=0` (No direction improves objective and is feasible)
- `\lambda_i* \left( g_i(x*)-b_i \right) = 0` (Complementary slackness)
- `\lambda_i* \ge 0` (Positive Lagrange multipliers)
The feasibility condition (1) applies to both equality and inequality constraints and is simply a statement that the constraints must not be violated at optimal conditions. The gradient condition (2) ensures that there is no feasible direction that could potentially improve the objective function. The last two condition (3 and 4) are only required with inequality constraints and enforce a positive Lagrange multiplier when the constraint is active (=0) and a zero Lagrange multiplier when the constraint is inactive (>0).
The necessary conditions for a constrained local optimum are called the Kuhn-Tucker Conditions, and these conditions play a very important role in constrained optimization theory and algorithm development.
The necessary conditions for a constrained local optimum are called the Karush Kuhn Tucker (KKT) Conditions, and these conditions play a very important role in constrained optimization theory and algorithm development. For an optimization problem:
$$\min_x f(x)$$ $$\mathrm{subject\;to}\;g_i(x)-b_i \ge 0 \quad i=1,\ldots,k$$ $$\quad\quad g_i(x)-b_i = 0 \quad i=k+1,\ldots,m$$
The four KKT conditions for optimality are
- `g_i(x)-b_i` is feasible
- `\grad f(x^*)-\sum_{i=1}^m \lambda_i^* \grad g_i\left(x^*\right)=0`
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Part 5: KKT Conditions for Dynamic Optimization
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