Karush-Kuhn-Tucker (KKT) Conditions

April 14, 2024, at 12:08 PM by 136.36.4.38 -

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Part 1: 5 Minute Tutorial on the KKT Conditions

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Part 1: Tutorial on the KKT Conditions

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Part 2: 5 Minute Tutorial with KKT Conditions and Inequality Constraints

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Part 2: KKT Conditions with Inequality Constraints

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Part 3: 5 Minute KKT Exercise with both Inequality and Equality Constraints

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Part 3: KKT Exercise with both Inequality and Equality Constraints

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Part 4: 5 Minute Application Exercise for the Optimal Volume of a Tank

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Part 4: Application Exercise for the Optimal Volume of a Tank

December 07, 2022, at 04:28 AM by 136.36.4.38 -

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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:

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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:

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1. Feasible Constraints

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1. Primal Feasibility: All constraints must be satisfied.

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2. No Feasible Descent

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2. No Feasible Descent: No possible objective improvement at the solution.

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3. Complementary Slackness

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3. Complementarity: The product of the Lagrange multipliers and the corresponding variables must be zero.

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4. Positive Lagrange Multipliers

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4. Dual Feasibility: The Lagrange multipliers associated with constraints have to be non-negative (zero or positive).

June 21, 2020, at 04:42 AM by 136.36.211.159 -

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October 28, 2019, at 01:11 PM by 136.36.211.159 -

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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).

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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 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).

December 20, 2018, at 03:05 PM by 173.117.150.72 -

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October 18, 2016, at 04:17 AM by 45.56.3.173 -

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There are four KKT conditions for optimality with the primal `x*` variable and dual `\lambda*` variable optimal values.

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There are four KKT conditions for optimal primal `(x)` and dual `(\lambda)` variables. The asterisk (*) denotes optimal values.

October 18, 2016, at 04:14 AM by 45.56.3.173 -

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$$\quad\quad\quad g_i(x)-b_i = 0 \quad i=k+1,\ldots,m$$

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$$\quad\quad\quad\quad\quad g_i(x)-b_i = 0 \quad i=k+1,\ldots,m$$

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$$g_i(x^*)-b_i$$

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$$g_i(x^*)-b_i \mathrm{\;is\;feasible}$$

October 18, 2016, at 04:13 AM by 45.56.3.173 -

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$$\mathrm{subject\;to}\;g_i(x)-b_i \ge 0 \quad i=1,\ldots,k$$ $$\quad\quad g_i(x)-b_i = 0 \quad i=k+1,\ldots,m$$

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$$\mathrm{subject\;to}\quad g_i(x)-b_i \ge 0 \quad i=1,\ldots,k$$ $$\quad\quad\quad g_i(x)-b_i = 0 \quad i=k+1,\ldots,m$$

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$$`\lambda_i^* \left( g_i(x^*)-b_i \right) = 0$$

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$$\lambda_i^* \left( g_i(x^*)-b_i \right) = 0$$

October 18, 2016, at 04:13 AM by 45.56.3.173 -

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$$\gradient f(x^*)-\sum_{i=1}^m \lambda_i^* \grad g_i\left(x^*\right)=0$$

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$$\nabla f(x^*)-\sum_{i=1}^m \lambda_i^* \nabla g_i\left(x^*\right)=0$$

October 18, 2016, at 04:12 AM by 45.56.3.173 -

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$$g_i(x*)-b_i$$

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$$g_i(x^*)-b_i$$

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$$\grad f(x*)-\sum_{i=1}^m \lambda_i^* \grad g_i\left(x*\right)=0$$

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$$\gradient f(x^*)-\sum_{i=1}^m \lambda_i^* \grad g_i\left(x^*\right)=0$$

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$$`\lambda_i* \left( g_i(x*)-b_i \right) = 0$$

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$$`\lambda_i^* \left( g_i(x^*)-b_i \right) = 0$$

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$$\lambda_i* \ge 0$$

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$$\lambda_i^* \ge 0$$

October 18, 2016, at 04:11 AM by 45.56.3.173 -

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`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)

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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

$$\lambda_i* \ge 0$$

October 18, 2016, at 04:08 AM by 45.56.3.173 -

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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`

to:

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).

October 18, 2016, at 03:57 AM by 45.56.3.173 -

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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.

to:

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`

June 15, 2015, at 02:14 PM by 45.56.3.184 -

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Part 5: KKT Conditions for Dynamic Optimization

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This assignment can be completed in groups of two. Additional guidelines on individual, collaborative, and group assignments are provided under the Expectations link.

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Design Optimization