Core Definitions in Linear Programming
Welcome to our comprehensive guide on LPP Definitions. In this lesson, we cover key geometric concepts that are essential to linear programming problems and the characterization of an optimal solution. You will explore hyperplanes, open and closed half spaces, lines, convex combinations, convex sets, and much more. Firstly, we explain each definition clearly and provide practical examples and proofs. This guide is designed in an engaging and accessible manner to help you master these fundamental concepts.
Hyperplane
Definition:
Let \(\pmb{\bar{c}} = [c_{1}, c_{2}, \dots, c_{n}]\) be a nonzero vector in \(n\)-dimensional Euclidean space \(\mathbb{E}^{n}\) and let \(z \in \mathbb{R}\). The set of points
\[
\pmb{\bar{x}} = [x_{1}, x_{2}, \dots, x_{n}]
\]
satisfying
\[
\pmb{\bar{c} \cdot \bar{x}} = c_{1}x_{1} + c_{2}x_{2} + \cdots + c_{n}x_{n} = z
\]
is called a hyperplane. This hyperplane effectively divides the space into regions, making it a fundamental tool for delineating feasible areas in linear programming.
Open Half Spaces
Definition:
With the same vector \(\pmb{\bar{c}}\) and scalar \(z\), an open half space consists of all points \(\pmb{\bar{x}}\) satisfying either
\[
\pmb{\bar{c} \cdot \bar{x}} < z \quad \text{or} \quad \pmb{\bar{c} \cdot \bar{x}} > z.
\]
Specifically, we define:
\[
H_{1} = \{\pmb{\bar{x}} : \pmb{\bar{c} \cdot \bar{x}} < z\} \quad \text{and} \quad H_{2} = \{\pmb{\bar{x}} : \pmb{\bar{c} \cdot \bar{x}} > z\}.
\]
These open half spaces help in identifying regions where optimal solutions might exist.
Closed Half Spaces
Definition:
A closed half space includes the boundary. Given \(\pmb{\bar{c}}\) and \(z\), the set of points satisfying
\[
\pmb{\bar{c} \cdot \bar{x}} \leq z \quad \text{or} \quad \pmb{\bar{c} \cdot \bar{x}} \geq z
\]
is defined as:
\[
H_{3} = \{\pmb{\bar{x}} : \pmb{\bar{c} \cdot \bar{x}} \leq z\} \quad \text{and} \quad H_{4} = \{\pmb{\bar{x}} : \pmb{\bar{c} \cdot \bar{x}} \geq z\}.
\]
Including the boundary is crucial for determining the limits of feasible solutions in LPP.
Line
Definition:
Let \(\pmb{\bar{\alpha}} = [\alpha_{1}, \alpha_{2}, \dots, \alpha_{n}]\) and \(\pmb{\bar{\beta}} = [\beta_{1}, \beta_{2}, \dots, \beta_{n}]\) be two distinct points in \(\mathbb{E}^{n}\). The set of all points
\[
\pmb{\bar{x}} = \lambda \pmb{\bar{\alpha}} + (1-\lambda) \pmb{\bar{\beta}}, \quad \forall \lambda \in \mathbb{R},
\]
represents the line that joins \(\pmb{\bar{\alpha}}\) and \(\pmb{\bar{\beta}}\). This concept is central to describing the boundaries within a linear programming model.
Line Segment
Definition:
Using the points \(\pmb{\bar{\alpha}}\) and \(\pmb{\bar{\beta}}\) in \(\mathbb{E}^{n}\), a line segment is defined as:
\[
\pmb{\bar{x}} = \lambda \pmb{\bar{\alpha}} + (1-\lambda) \pmb{\bar{\beta}}, \quad \forall \lambda \in [0, 1].
\]
By restricting \(\lambda\) to the interval \([0, 1]\), the line segment remains bounded, which is vital when representing finite feasible regions.
Convex Combinations
Definition:
Suppose you have \(r\) points \(\pmb{\bar{x}_{1}}, \pmb{\bar{x}_{2}}, \dots, \pmb{\bar{x}_{r}}\) in \(\mathbb{E}^{n}\). A point \(\pmb{\bar{x}}\) is a convex combination of these points if it can be expressed as:
\[
\pmb{\bar{x}} = \lambda_{1}\pmb{\bar{x}_{1}} + \lambda_{2}\pmb{\bar{x}_{2}} + \cdots + \lambda_{r}\pmb{\bar{x}_{r}},
\]
with the conditions
\[
\lambda_{1} + \lambda_{2} + \cdots + \lambda_{r} = 1 \quad \text{and} \quad \lambda_{i} \geq 0 \text{ for each } i.
\]
Convex combinations are key to constructing the feasible set in linear programming.
Convex Sets
Definition:
A set \(X \subset \mathbb{E}^{n}\) is a convex set if, for any two points \(\pmb{\bar{x}_{1}}\) and \(\pmb{\bar{x}_{2}}\) in \(X\), the line segment joining them lies entirely within \(X\). In formal terms, for every \(\lambda \in [0, 1]\),
\[
\lambda \pmb{\bar{x}_{1}} + (1-\lambda) \pmb{\bar{x}_{2}} \in X.
\]
Convex sets are important because they guarantee that any local optimal solution is also globally optimal in LPP.
Example: Proving a Convex Set
-
Problem: Prove that
\[
X = \{\pmb{\bar{x}} : |\pmb{\bar{x}}| \leq 2\}
\]
is a convex set.
Solution:
Let \(\pmb{\bar{x}_{1}}, \pmb{\bar{x}_{2}} \in X\) so that \(|\pmb{\bar{x}_{1}}| \leq 2\) and \(|\pmb{\bar{x}_{2}}| \leq 2\). Firstly, consider any point on the line segment between them: \[ \pmb{\bar{x}} = \lambda \pmb{\bar{x}_{1}} + (1-\lambda) \pmb{\bar{x}_{2}}, \quad \lambda \in [0,1]. \] Next, by applying the triangle inequality we have: \[ |\pmb{\bar{x}}| \leq \lambda |\pmb{\bar{x}_{1}}| + (1-\lambda)|\pmb{\bar{x}_{2}}| \leq 2\lambda + 2(1-\lambda) = 2. \] Thus, \(\pmb{\bar{x}} \in X\), which confirms that \(X\) is convex.
Extreme Points
Definition:
In a convex set \(X \subset \mathbb{E}^{n}\), a point \(\pmb{\bar{x}} \in X\) is an extreme point if it cannot be represented as a convex combination of any two distinct points in \(X\). Extreme points are of great interest in linear programming since optimal solutions often occur at these points.
Convex Hull
Definition:
The convex hull of a set \(X \subset \mathbb{E}^{n}\) is the smallest convex set that contains \(X\). It is defined as the set of all convex combinations of points in \(X\):
\[
\text{conv}(X) = \left\{\sum_{i=1}^{r} \lambda_{i} \pmb{\bar{x}_{i}} : \pmb{\bar{x}_{i}} \in X,\; \lambda_{i} \geq 0,\; \sum_{i=1}^{r} \lambda_{i} = 1\right\}.
\]
This concept plays a crucial role in determining the boundaries of feasible regions.
Convex Polyhedron
Definition:
If \(X\) is a finite set in \(\mathbb{E}^{n}\), the convex polyhedron generated by \(X\) is the set of all convex combinations of its points:
\[
P = \left\{\sum_{i=1}^{r} \lambda_{i} \pmb{\bar{x}_{i}} : \pmb{\bar{x}_{i}} \in X,\; \lambda_{i} \geq 0,\; \sum_{i=1}^{r} \lambda_{i} = 1\right\}.
\]
Convex polyhedra are central to linear programming, as they define the feasible region over which optimization occurs.
Conclusion
In conclusion, mastering these LPP Definitions is essential for solving linear programming problems. By understanding hyperplanes, half spaces, lines, convex combinations, and convex sets, you equip yourself with the tools necessary to characterize and find optimal solutions. Additionally, these definitions lay the groundwork for advanced optimization techniques and further exploration in linear programming. We hope this guide has provided clear insights into the geometric foundations of LPP.
Related Docs
FAQs
Linear Programming Problems
- What is a linear programming problem?
A linear programming problem is an optimization problem where the objective is to maximize or minimize a linear function subject to a set of linear constraints.
- What are the main components of a linear programming problem?
The main components include:
- Objective Function: The function to be maximized or minimized.
- Decision Variables: The variables whose values determine the outcome.
- Constraints: Linear inequalities or equations that restrict the values of the decision variables.
- Feasible Region: The set of all solutions that satisfy the constraints.
- What is the objective function in linear programming?
The objective function is a linear equation representing the goal of the problem. It defines what needs to be maximized or minimized, such as profit, cost, or time.
- What is the objective function in linear programming?
The objective function is a linear equation representing the goal of the problem. It defines what needs to be maximized or minimized, such as profit, cost, or time.
- What are constraints in linear programming?
Constraints are the linear inequalities or equations that specify the limitations or requirements of the problem. They define the boundaries of the feasible region.
- What are constraints in linear programming?
Constraints are the linear inequalities or equations that specify the limitations or requirements of the problem. They define the boundaries of the feasible region.
- What is the feasible region?
The feasible region is the set of all possible points (solutions) that satisfy all of the problem's constraints. In a two-variable problem, it is typically represented as a polygon on a graph.
- What is the Simplex method?
The Simplex method is an algorithm used to solve linear programming problems. It systematically examines the vertices of the feasible region to find the optimal solution.
- How are linear programming problems solved?
These problems can be solved using various methods such as the Simplex algorithm, Interior-Point methods, or graphical methods (for problems with two variables). The choice of method depends on the size and complexity of the problem.
- What is duality in linear programming?
Duality refers to the concept that every linear programming problem (known as the primal problem) has an associated dual problem. The solutions of the dual provide insights into the primal problem, and under certain conditions, their optimal values are equal.
- What are some common applications of linear programming?
Linear programming is widely used in various fields such as:
- Operations Research
- Economics
- Supply Chain Management
- Production Planning
- Transportation and Logistics
- What is sensitivity analysis in linear programming?
Sensitivity analysis examines how changes in the parameters of a linear programming problem (such as coefficients in the objective function or the right-hand side values of constraints) affect the optimal solution, helping decision-makers understand the robustness of the solution.
-
Vidyasagar University- Quesitions Papers
- Zoology Honours- Vidyasagar University- Quesitions Papers
- Physiology Honours- Vidyasagar University- Quesitions Papers
- Physics Honours- Vidyasagar University- Quesitions Papers
- Nutrition Honours- Vidyasagar University- Quesitions Papers
- Geography Honours- Vidyasagar University- Quesitions Papers
- Sociology Honours- Vidyasagar University- Quesitions Papers
- Sanskrit Honours- Vidyasagar University- Quesitions Papers
- Political Science Honours- Vidyasagar University- Quesitions Papers
- Philosophy Honours- Vidyasagar University- Quesitions Papers
- Music Honours- Vidyasagar University- Quesitions Papers