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Betrachten Sie im Folgenden den durch die Ungleichungen $$ \begin{array}{r} x_{1}+x_{2} \leq 5 \\ -x_{1}+x_{2} \leq 1 \\ x_{1}, x_{2} \geq 0 \end{array} $$ definierten Polyeder und die Zielfunktion \(z=c^{\mathrm{T}} \boldsymbol{x}\) mit dem zugeh枚rigen, von den beiden Gr枚脽en \(r>0\) und \(\alpha \in[0,2 \pi\) [ abh盲ngigen Zielfunktionsvektor \(\boldsymbol{c}=\boldsymbol{c}(r, \alpha)=(r \cos \alpha, r \sin \alpha)^{\mathrm{T}}\). (a) Bestimmen Sie grafisch die optimalen Ecken des Optimierungsproblems f眉r \(r=1, \alpha=\frac{3 \pi}{8}\) sowie f眉r \(r=2\) und \(\alpha=\frac{5 \pi}{8}\) (b) Bestimmen Sie die Menge aller \(r>0\) und \(\alpha \in[0,2 \pi[\), f眉r die die Ecke \((2,3)^{\mathrm{T}}\) des Polyeders eine optimale L枚sung des Optimierungsproblems ist. Gehen Sie dazu zun盲chst grafisch vor und beweisen Sie anschlieBend Ihre Vermutung mathematisch. (c) Die Nebenbedingungen, f眉r die in einem Punkt eines durch Ungleichungen gegebenen Polyeders sogar Gleichheit gilt, bezeichnet man als die in diesem Punkt aktiven Nebenbedingungen. Den zu einer Ungleichung \(\boldsymbol{a}^{\mathrm{T}} \boldsymbol{x} \leq b\) geh枚rigen Vektor \(\boldsymbol{a}\) nennt man den Gradienten dieser Ungleichung. Betrachten Sie nun den von den Gradienten der in der Ecke \((2,3)^{\mathrm{T}}\) des Polyeders aktiven Nebenbedingungen aufgespannten Kegel, das hei脽t die Menge $$ K=\left\\{\lambda(1,1)^{\mathrm{T}}+\mu(-1,1)^{\mathrm{T}} \mid \lambda, \mu \geq 0\right\\} $$ F眉r welche \(r>0\) und \(\alpha \in[0,2 \pi[\) gilt \(c(r, \alpha) \in K\) ? Beweisen Sie Ihre Aussage!

Step by step solution

01

Plot the inequalities of the polyhedron

First, we need to plot the inequalities defined on the 鈩漗2 plane: $$ \begin{array}{r} x_{1}+x_{2} \leq 5 \\\ -x_{1}+x_{2} \leq 1 \\\ x_{1}, x_{2} \geq 0 \end{array} $$ We know that all the points satisfying these inequalities will lie within the polyhedron. This can be done using any graphing tool of your choice.

02

Find the corner points of the polyhedron

To find the corner points, we need to determine the intersections of the equations forming the inequalities above. Calculations turn out as follows: Intersection of x1 + x2 = 5, -x1 + x2 = 1: x1 = 2 x2 = 3 Hence, corner point A is (2,3). Intersection of x1 + x2 = 5, x1 = 0: x1 = 0 x2 = 5 Thus, corner point B is (0,5). Intersection of -x1 + x2 = 1, x1 = 0: x1 = 0 x2 = 1 Therefore, corner point C is (0,1). Intersection of -x1 + x2 = 1, x2 = 0: x1 = 1 x2 = 0 So, corner point D is (1,0).

03

Step 3a: Evaluate the function z at each corner for 伪 = 3蟺/8 and r = 1

Now, we want to find the optimal corners of the polyhedron for the values of 伪 = 3蟺/8 and r = 1. To do this, we just need to plug in the given 伪 and r values into c and then plug c into our function z. After evaluating each corner point for the given values, compare the values of z at each corner to determine the optimal one.

04

Step 3b: Evaluate the function z at each corner for 伪 = 5蟺/8 and r = 2

Repeat the same process for the values of 伪 = 5蟺/8 and r = 2.

05

Determine the range of r and 伪 for which point (2,3) is an optimal solution

To find what values of r and 伪 will make the corner (2,3) an optimal solution, perform a graphical investigation on the domain (0,2蟺) for 伪, and positive r values. Then, prove your observations mathematically.

06

Analyze the conic area K spanned by the gradients of active constraints

Consider the conic area K given by the gradients of active constraints at point (2,3): $$ K=\left\\{\lambda(1,1)^{\mathrm{T}}+\mu(-1,1)^{\mathrm{T}} \mid \lambda, \mu \geq 0\right\\} $$ For which values of r and 伪, can we say that c(r, 伪) 鈭� K? Determine these values and justify your answer for any r > 0 and 伪 鈭� [0, 2蟺).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Polyhedrons in Linear Optimization

A polyhedron is a geometric figure with flat faces and straight edges, prevalent in linear optimization problems. When studying linear optimization, a polyhedron often refers to the feasible region defined by a set of linear inequalities.

In the exercise, the polyhedron in question is confined within the plane by three inequalities:

• \( x_{1} + x_{2} \leq 5 \)
• \( -x_{1} + x_{2} \leq 1 \)
• \( x_{1}, x_{2} \geq 0 \)

Visualizing this polyhedron helps students understand the set of possible solutions. It's crucial to conceptualize each face representing a constraint and the corners, or vertices, where these faces meet. These vertices often hold the key to finding the optimal solution in linear optimization problems.

Finding the Optimal Solution

The optimal solution in linear optimization is the best possible outcome within the model's constraints. It is usually a point in the polyhedron that maximizes or minimizes the objective function.

In our exercise, the objective function is \( z = c^{T} \boldsymbol{x} \) with the variable vector \( \boldsymbol{c} \). For specific values of \( r \) and \( \alpha \), we evaluate each vertex of the polyhedron to determine which offers the highest or lowest value of the objective function. These values are deciphered graphically, which is an accessible method for most students, and then mathematic proofs solidify these findings.

Exercise Improvement Tip:

Include visual aids, like graphs, showing the polyhedron and how different objective function values change over the region, to enhance concept clarity.

Active Constraints and Their Role

Active constraints are equations that directly impact the solution of an optimization problem. At any point in the feasible region, especially at the vertices, some inequality constraints become equalities - these are the active constraints.

In the provided exercise, identifying which constraints are active at the vertex \((2,3)\) can determine the gradient vectors of these constraints. Recognizing and visualizing the active constraints allow students to better understand the structure of the problem and how alterations to these constraints can influence the optimal solution.

Exercise Improvement Tip:

Suggest verifying the activity of a constraint by plugging the coordinates of the vertex into the original inequalities.

Gradient Vectors and Their Significance

Gradient vectors point towards the direction of the greatest increase of a function. In the context of linear optimization, the gradient vectors of the constraints can create a cone that encapsulates the directions in which the objective function can increase.

The exercise asks to analyze the cone formed by the gradient vectors of the active constraints at the point \((2,3)\) to investigate for which values of \( r \) and \( \alpha \), the objective function vector \( c(r, \alpha) \) falls within this cone. This understanding is integral as it relates to the sensitivity of the optimal solution to changes in the objective function's direction.

Exercise Improvement Tip:

Encourage students to visualize gradient vectors using available software tools to comprehend the concept of directional increases in functions and their impact on the solution.