Show that the branch-and-bound method will always branch if seven or less variables have been branched on. (In fact, most of the branches have almost all the variables set.) This shows that branch-and-bound can take exponential time. For this problem, many solvers would catch what is happening in preprocessing and find the optimal solution almost immediately.
Show that the branch-and-bound method will always branch if seven or less variables have been branched on. (In fact, most of the branches have almost all the variables set.) This shows that branch-and-bound can take exponential time. For this problem, many solvers would catch what is happening in preprocessing and find the optimal solution almost immediately.
Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
Problem 1PE
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Question
Solve the following problem and find the optimal solution.
![**Optimization Problem and Branch-and-Bound Method**
Consider the problem:
**Objective:**
Minimize: \( y \)
**Subject to:**
\[ 2x_1 + 2x_2 + \cdots + 2x_{15} + y \leq 15 \]
**Variable Constraints:**
\[ x_1, x_2, \ldots, x_{15}, y \in \{0, 1\} \]
**Problem Explanation:**
Show that the branch-and-bound method will always branch if seven or fewer variables have been branched on. (In fact, most of the branches have almost all the variables set.) This illustrates that branch-and-bound can take exponential time. For this problem, many solvers would quickly recognize the situation during preprocessing and find the optimal solution almost immediately.
**Discussion:**
The challenge lies in handling the complexity of the branch-and-bound method when applied to this type of problem. The method can become computationally intensive when faced with a higher number of variables requiring branching. The problem demonstrates how solvers can optimize performance by recognizing patterns or simplifying conditions early in the process.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F65a28526-6c14-4460-a6e3-54f91e103386%2F5c24a2aa-c49a-4459-8410-22d5a901a839%2F6t1ylsi_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Optimization Problem and Branch-and-Bound Method**
Consider the problem:
**Objective:**
Minimize: \( y \)
**Subject to:**
\[ 2x_1 + 2x_2 + \cdots + 2x_{15} + y \leq 15 \]
**Variable Constraints:**
\[ x_1, x_2, \ldots, x_{15}, y \in \{0, 1\} \]
**Problem Explanation:**
Show that the branch-and-bound method will always branch if seven or fewer variables have been branched on. (In fact, most of the branches have almost all the variables set.) This illustrates that branch-and-bound can take exponential time. For this problem, many solvers would quickly recognize the situation during preprocessing and find the optimal solution almost immediately.
**Discussion:**
The challenge lies in handling the complexity of the branch-and-bound method when applied to this type of problem. The method can become computationally intensive when faced with a higher number of variables requiring branching. The problem demonstrates how solvers can optimize performance by recognizing patterns or simplifying conditions early in the process.
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