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Questions and Answers
The region described by $X = {x = (x_1, x_2) ∈ R^2 : Ax ≤ b}$, where $A = \begin{pmatrix} 1 & 1 \ 2 & -1 \ 0 & 1 \end{pmatrix}$ and $b = \begin{pmatrix} 4 \ 6 \ 2 \end{pmatrix}$, is both non-empty and bounded.
The region described by $X = {x = (x_1, x_2) ∈ R^2 : Ax ≤ b}$, where $A = \begin{pmatrix} 1 & 1 \ 2 & -1 \ 0 & 1 \end{pmatrix}$ and $b = \begin{pmatrix} 4 \ 6 \ 2 \end{pmatrix}$, is both non-empty and bounded.
False (B)
If a set $X = {x = (x_1, x_2) ∈ R^2 : x_1 + x_2 ≥ 2, x_1 ≥ 0, x_2 ≥ 0}$, then a direction of $X$ is defined as a vector $d$ such that for all $x ∈ X$ and all $λ ≥ 0$, the vector $x + λd$ is not necessarily an element of $X$.
If a set $X = {x = (x_1, x_2) ∈ R^2 : x_1 + x_2 ≥ 2, x_1 ≥ 0, x_2 ≥ 0}$, then a direction of $X$ is defined as a vector $d$ such that for all $x ∈ X$ and all $λ ≥ 0$, the vector $x + λd$ is not necessarily an element of $X$.
False (B)
Given $X = {x = (x_1, x_2) ∈ R^2 : -x_1 + x_2 ≤ 2; -x_1 + 2x_2 ≤ 6; x_1 ≥ 0, x_2 ≥ 0}$, the extreme points of $X$ are $(0, 0)$, $(0, 3)$, and $(2, 4)$.
Given $X = {x = (x_1, x_2) ∈ R^2 : -x_1 + x_2 ≤ 2; -x_1 + 2x_2 ≤ 6; x_1 ≥ 0, x_2 ≥ 0}$, the extreme points of $X$ are $(0, 0)$, $(0, 3)$, and $(2, 4)$.
True (A)
For the optimization problem $\text{max } z = 3x_1 + x_2$ subject to $-x_1 + 2x_2 ≤ 0$ and $x_1, x_2 ≥ 0$, the optimal solution is unbounded.
For the optimization problem $\text{max } z = 3x_1 + x_2$ subject to $-x_1 + 2x_2 ≤ 0$ and $x_1, x_2 ≥ 0$, the optimal solution is unbounded.
The set $X = {x = (x_1, x_2) ∈ R^2 : -x_1 + x_2 ≤ 2; -x_1 + 2x_2 ≤ 6; x_1 ≥ 0, x_2 ≥ 0}$ is a convex set because it is defined by a system of linear inequalities.
The set $X = {x = (x_1, x_2) ∈ R^2 : -x_1 + x_2 ≤ 2; -x_1 + 2x_2 ≤ 6; x_1 ≥ 0, x_2 ≥ 0}$ is a convex set because it is defined by a system of linear inequalities.
The set of extreme directions for $X = {x = (x_1, x_2) ∈ R^2 : x_1 + x_2 ≥ 2, x_1 ≥ 0, x_2 ≥ 0}$ consists of two directions: $d_1 = (1, 0)$ and $d_2 = (0, -1)$.
The set of extreme directions for $X = {x = (x_1, x_2) ∈ R^2 : x_1 + x_2 ≥ 2, x_1 ≥ 0, x_2 ≥ 0}$ consists of two directions: $d_1 = (1, 0)$ and $d_2 = (0, -1)$.
In the requirement space for the problem $\text{max } z = 3x_1 + x_2$ subject to $-x_1 + 2x_2 ≤ 0$ and $x_1, x_2 ≥ 0$, feasibility is represented by the set of points where the constraints intersect, graphically showing which combinations of resources satisfy the requirements.
In the requirement space for the problem $\text{max } z = 3x_1 + x_2$ subject to $-x_1 + 2x_2 ≤ 0$ and $x_1, x_2 ≥ 0$, feasibility is represented by the set of points where the constraints intersect, graphically showing which combinations of resources satisfy the requirements.
If $X = {x = (x_1 , x_2 ) ∈ R^2 : Ax ≤ b}$ where $A = \begin{pmatrix} -1 & 0 \ 0 & -1 \ 2 & 3 \ 1 & -1 \end{pmatrix}$ and $b = \begin{pmatrix} 0 \ 0 \ 6 \ 5 \end{pmatrix}$, then the region described by $X$ is empty.
If $X = {x = (x_1 , x_2 ) ∈ R^2 : Ax ≤ b}$ where $A = \begin{pmatrix} -1 & 0 \ 0 & -1 \ 2 & 3 \ 1 & -1 \end{pmatrix}$ and $b = \begin{pmatrix} 0 \ 0 \ 6 \ 5 \end{pmatrix}$, then the region described by $X$ is empty.
Flashcards
Region described by X = {x : Ax ≤ b}
Region described by X = {x : Ax ≤ b}
A set of points x that satisfy a set of linear inequalities Ax ≤ b.
Empty Region
Empty Region
A region is empty if there are no points that satisfy all the inequalities.
Bounded Region
Bounded Region
A region is bounded if it can be enclosed within a circle of finite radius.
Direction of a Set
Direction of a Set
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Extreme Direction
Extreme Direction
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Convex Set
Convex Set
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Extreme Point
Extreme Point
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Sketching Feasible Region
Sketching Feasible Region
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Study Notes
- The problems relate to mathematical programming homework.
Problem 1
- Requires sketching regions described by the set X = {(x1, x2) ∈ R² : Ax ≤ b} for various matrices A and vectors b.
- Determine if the regions are empty and/or bounded.
Problem 1(a)
- A = [[1, 1], [0, 1]], b = [[4], [6], [2]]
Problem 1(b)
- A = [[-1, 0], [-1, 0], [2, 3], [1, -1]], b = [[0], [0], [6], [5]]
Problem 1(c)
- A = [[1, 1], [-1, -2], [-1, 0]], b = [[5], [-12], [0]]
Problem 2
- Given X = {x = (x1, x2) ∈ R² : x1 + x2 ≥ 2, x1 ≥ 0, x2 ≥ 0}.
- The task is to describe how a direction of X is defined, find all extreme directions of X, and determine all directions of X.
Problem 3
- Given X = {x = (x1, x2) ∈ R² : -x1 + x2 ≤ 2; -x1 + 2x2 ≤ 6; x1 ≥ 0, x2 ≥ 0}.
- Need to show that X is a convex set and find all extreme points and extreme directions of X.
Problem 4
- Considers the optimization problem: maximize z = 3x1 + x2 subject to -x1 + 2x2 ≤ 0 and x1, x2 ≥ 0.
- Requires sketching the feasible region in the (x1, x2) space, solving the problem geometrically, and drawing the requirement space and interpreting feasibility.
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