Optimization - Nicolas Ortiz

sketchfab

Here is your optimization homework: 1. Write a C program to implement Dijkstra's algorithm. The program should take as input a graph represented by an adjacency matrix and output the shortest distance from a given source node to all other nodes in the graph. 2. Solve the following linear programming problem using the simplex method: Maximize Z = 5x + 7y subject to x + y <= 4, x >= 0, y >= 0. 3. Prove that every undirected graph with a cycle has at least two nodes of odd degree. 4. Solve the following recurrence relation using iteration: T(n) = 2T(n/2) + n^2. 5. Write a Python program to implement a stack using an array. The program should include functions for push, pop, and peek operations. 6. Find all possible solutions to the following system of linear equations: x - y + z = 0, x + y - z = 0, 2x - 3y + z = 0. 7. Write a Java program to implement a priority queue using an array. The program should include functions for insert and delete operations. 8. Prove that every connected graph with no cycles has exactly one spanning tree. 9. Solve the following optimization problem: Minimize Z = x^2 + y^2 subject to x^2 + y^2 <= 4. 10. Write a C++ program to implement a queue using an array. The program should include functions for enqueue and dequeue operations.