Question

Is the Kuhn-Tucker sufficiency theorem applicable to: (a) Maximize \(\quad \pi=x_{1}\) \\[ \text { subject to } \quad x_{1}^{2}+x_{3}^{2} \leq 1 \\] and \\[ x_{1}, x_{2} \geq 0 \\] (b) Minimize \(\quad C=\left(x_{1}-3\right)^{2}+\left(x_{2}-4\right)^{2}\) \\[ \text { subject to } \quad x_{1}+x_{2} \geq 4 \\] and \\[ x_{1}, x_{2} \geq 0 \\] (c) Minimize \(\quad C=2 x_{1}+x_{2}\) \\[ \text { subject to } \quad x_{1}^{2}-4 x_{1}+x_{2} \geq 0 \\] and \\[ x_{1}, x_{2} \geq 0 \\]

Short answer

(a) Applicable, (b) Applicable, (c) Not generally applicable without further assumptions.

Step-by-step solution

Understanding the Objective Functions and Constraints

In part (a), the aim is to maximize the function  \( \pi = x_1 \)  with the constraints \( x_1^2 + x_3^2 \leq 1 \) and \( x_1, x_2 \geq 0 \). In part (b), the aim is to minimize the cost function \( C = (x_1 - 3)^2 + (x_2 - 4)^2 \) subject to the constraints \( x_1 + x_2 \geq 4 \) and \( x_1, x_2 \geq 0 \). Part (c) involves minimizing the cost function \( C = 2x_1 + x_2 \) with the constraint \( x_1^2 - 4x_1 + x_2 \geq 0 \) and \( x_1, x_2 \geq 0 \).

Applying the Kuhn-Tucker Theorem Criteria

The Kuhn-Tucker conditions apply to optimization problems that involve differentiable functions with inequality constraints. These conditions are particularly used when both the objective function and the constraint functions are continuously differentiable and the constraint region is convex.

Convexity and Differentiability Analysis (Part a)

For part (a), the function \( \pi = x_1 \) is linear, and the constraint \( x_1^2 + x_3^2 \leq 1 \) describes a convex set (a circle or ellipse). The partial derivatives exist, and the constraints form a convex region. Therefore, the Kuhn-Tucker sufficiency theorem is applicable.

Convexity and Differentiability Analysis (Part b)

In part (b), the objective function \( C = (x_1 - 3)^2 + (x_2 - 4)^2 \) is a quadratic function, which is convex. The constraint \( x_1 + x_2 \geq 4 \) is linear, describing a half-space, which is a convex set. The functions are continuously differentiable, so the Kuhn-Tucker sufficiency theorem is applicable.

Convexity and Differentiability Analysis (Part c)

In part (c), the objective function \( C = 2x_1 + x_2 \) is linear. However, the constraint \( x_1^2 - 4x_1 + x_2 \geq 0 \) involves a quadratic term, and analyzing the feasible region formed by this constraint is necessary. Upon analysis, it can be found that the constraint does not necessarily describe a convex set throughout its domain. Therefore, the Kuhn-Tucker sufficiency theorem might not generally be applicable without additional assumptions about the convexity of the feasible region.

Key concepts

Convexity

Convexity is a fundamental concept in understanding optimization problems and their solutions. If the objective function and constraints form a convex set, it simplifies the problem substantially. But what does convexity really mean? A set is considered convex if, for any two points within the set, the straight line connecting these points lies entirely within the set. Convex sets resemble nicely rounded bodies without any indentations or holes. A common example is a line segment between two points that solely remains inside the circle or ellipse it belongs to.
A function is convex if its graph lies below any straight line segment connecting two points on the graph. In mathematical terms, a differentiable function is convex if its second derivative is non-negative across its entire domain. In a more simple sense, think of it as if the function curves upwards, like a bowl. Convexity is crucial in optimization as it ensures that local minima are also global minima, simplifying solution processes like applying the Kuhn-Tucker conditions.

Differentiability

Differentiability plays a critical role in optimization, particularly when applying the Kuhn-Tucker sufficiency theorem. Put simply, a function is said to be differentiable if it has a derivative at each point in its domain. The derivative offers a way to understand how the function behaves and changes at any given point. Think of it as the slope of the tangent line to the curve at a specific point.
When tackling optimization problems, differentiability allows us to apply calculus-based techniques, such as finding critical points where the derivative equals zero. This helps identify potential extremes, such as minima or maxima, in the objective function. The Kuhn-Tucker theorem demands that both the objective function and the constraint functions should be continuously differentiable. This smoothness ensures that the applied optimization logic holds throughout the function's domain.

Optimization Problems

Optimization problems involve finding the best solution from a set of feasible solutions, given constraints. These problems can take various forms, such as maximizing a profit or minimizing a cost. Generally, they consist of an objective function and possible constraints that define the solution space.
In these problems, constraints can be equations or inequalities that the solution must satisfy. The objective function represents the value that is to be optimized. For instance, you might have an objective function representing cost, which you intend to reduce while respecting budget limits (constraints).
The Kuhn-Tucker conditions are a set of requirements under which a solution to a constrained optimization problem is optimal. These conditions extend the logic of simple, unconstrained optimization to situations where inequality constraints are present. Understanding the nature of the objective function and constraints—whether they are linear, convex, or differentiable—serves as the gateway to successfully applying strategies like the Kuhn-Tucker theorem.