Question

\\[ \begin{array}{ll} \text { Minimize } & C=x_{1} \\ \text { subject to } & -x_{2}-\left(1-x_{1}\right)^{3} \geq 0 \end{array} \\] and \\[ x_{1}, x_{2} \geq 0 \\] Show that \((a)\) the optimal solution \(\left(x_{1}^{*}, x_{2}^{*}\right)=(1,0)\) does not satisfy the Kuhn-Tucker conditions, but \((b)\) by introducing a new muttiplier \(\lambda_{0} \geq 0,\) and modifying the Lagrangian function (13.15) to the form \\[ Z_{0}=\lambda_{0} f\left(x_{1}, x_{2}, \ldots, x_{n}\right)+\sum_{j=1}^{m} \lambda_{1}\left[r_{i}-g^{j}\left(x_{1}, x_{2}, \ldots, x_{n}\right)\right] \\] the Kuhn-Tucker conditions can be satisfied at (1,0) . (Note: The Kuhn-Tucker conditions on the multipliers extend to only \(\left.\dot{\lambda}_{1}, \ldots, \lambda_{m}, \text { but not to } \lambda_{0} .\right)\)

Short answer

At \((1,0)\), the Kuhn-Tucker conditions require a new Lagrangian with \(\lambda_0\) to be satisfied.

Step-by-step solution

Define the Objective and Constraint Functions

Here, we need to minimize the function \(C = x_1\), subject to the constraint \(-x_2 - (1-x_1)^3 \geq 0\) with non-negativity conditions \(x_1, x_2 \geq 0\). Identify the objective function as \( f(x_1, x_2) = x_1 \) and the constraint as \( g(x_1, x_2) = -x_2 - (1-x_1)^3 \geq 0 \).

Check Kuhn-Tucker Conditions with Existing Multipliers

The Kuhn-Tucker conditions require a Lagrangian with multipliers \( \lambda_1 \geq 0 \) such that \( \lambda_1 g(x_1, x_2) = 0 \) and the gradient condition \( abla f(x_1, x_2) + \lambda_1 abla g(x_1, x_2) = 0 \). Check these at the proposed solution \((x_1^*, x_2^*) = (1, 0)\).

Gradient of Objective and Constraint Functions

Compute the gradients: \( abla f = (1, 0) \) and \( abla g = (3(1-x_1)^2, -1) \). At \( (1, 0) \), the gradient of \( g \) simplifies to \( (0, -1) \).

Kuhn-Tucker Gradient Condition Check

The Kuhn-Tucker condition becomes \( (1, 0) + \lambda_1 (0, -1) = (0, 0) \). Thus, \( 1=0 \) and \( -\lambda_1 = 0 \), making \( \lambda_1 = 0 \), contradictory.

Modify the Lagrangian with \(\lambda_0\)

Introduce \( \lambda_0 \geq 0 \) and redefine the Lagrangian as \( Z_0 = \lambda_0 x_1 - \lambda_1 (x_2 + (1-x_1)^3) \). Ensure \(\lambda_1\) triggers active constraints.

Revised Gradient Condition with \(\lambda_0\)

Apply the gradient condition: \( \lambda_0 (1, 0) - \lambda_1 (3(1-x_1)^2, 1) = (0, 0) \). At \((1, 0)\), it simplifies to \( \lambda_0 = 0 \) if \(\lambda_1 eq 0\).

Verify the Solution Satisfies Revised Conditions

With \(\lambda_1 = 1\), check \( \lambda_1 g(1, 0) = -0^3 = 0 \) satisfied. The multiplier \(\lambda_0 = 0\) aligns with allowing \(\lambda_1\) to solely satisfy conditions.

Key concepts

Lagrangian Function

The Lagrangian function is a vital tool in optimization problems, particularly those involving constraints. While dealing with constrained optimization, we use the Lagrangian function to incorporate constraints into the objective function.
This allows us to deal with a single equation rather than multiple. Consider the Lagrangian for our problem, defined initially as:\[ L(x_1, x_2, \lambda_1) = x_1 - \lambda_1(-x_2 - (1-x_1)^3) \]Here, we introduce a constraint multiplier, \(\lambda_1\), to handle the constraint. The main advantage of using the Lagrangian is that it provides a way to search for optimal solutions by finding conditions under which the gradient of the Lagrangian is zero. This simplification makes it easier to identify critical points of the optimization problem.

Optimization Problem

Optimization problems seek to find the best solution, usually the maximum or minimum, under given conditions. In this exercise, we aim to minimize the function \(C = x_1\), subject to the constraints given.
The optimization problem here includes a primary function, or objective function, \(f(x_1, x_2) = x_1\), whose minimum value we want to achieve, and a constraint given by \(g(x_1,x_2) = -x_2 - (1-x_1)^3 \geq 0\).
Solving optimization problems with constraints involves finding a balance or a trade-off between the objective you're looking to optimize and the constraints you must satisfy. This is a foundational skill in many fields like economics, engineering, and operations research, emphasizing the necessity to make decisions that best align with conditions set by external factors.

Constraint Function

Constraints are conditions or limits imposed on the optimization problem. In this example, the constraint function is \(g(x_1, x_2) = -x_2 - (1-x_1)^3 \geq 0\). This constraint limits the feasible region in which the solution can be found.
Constraints shape the solution space, such that only the solutions that satisfy these constraints are considered viable solutions. The constant \(x_1, x_2 \geq 0\) further confines the solution space.
Constraints can be equality or inequality, and in this case, it's an inequality. Handling such constraints often requires leveraging methods like the KKT (Karush-Kuhn-Tucker) conditions, which support navigating complex solution spaces and finding optimal solutions that also respect the imposed constraints.

Multiplier Method

The multiplier method is a technique used to incorporate constraints into an objective function, allowing us to solve constrained optimization problems. The multipliers, often denoted by the Greek letter \(\lambda\), act as weights that penalize the violation of the constraints.
In our specific problem, we noticed that initially, the Kuhn-Tucker conditions were not satisfied, as indicated by calculations leading to contradictions. Thus, introducing a new multiplier \(\lambda_0\) was necessary. Revising the Lagrangian to:\[ Z_0 = \lambda_0 x_1 - \lambda_1(x_2 + (1-x_1)^3) \]The multiplier \(\lambda_0\) helps reformulate the conditions and achieve a feasible solution.
The primary application of the multiplier method is in finding optimal solutions in complex, constrained environments. It also reveals whether or not the current set of solutions can be optimized further or if adjustments to constraints are necessary to find viable solutions. This approach is especially useful in economics and other decision-making scenarios, where constraints are common.