Advanced Algorithms (COMPSCI 224), Lecture 16

TL;DR

The Simplex algorithm iteratively moves from vertex to vertex to find the optimal solution, while the Interior Point algorithm stays inside the polytope and gradually moves towards the optimal vertex.

Transcript

oh okay I'm gonna get started let me wait so the cameras on okay good so last time so okay so today let me say today we're going to finish simplex and then and then I'm going to say things about strong duality and complementary slackness you'll see what I mean what that means I'm going to sort of tell so the simplex algorithm unfortunately there ar... Read More

Key Insights

• 💠 The Simplex algorithm moves from vertex to vertex in a polytope to find the optimal solution, while the Interior Point algorithm stays inside the polytope and gradually moves towards the optimal vertex.
• 😒 The Ellipsoid algorithm uses ellipsoids to check the feasibility of a polytope and can solve exponentially sized linear programs in polynomial time.
• ❓ Complementary slackness is a condition in linear programming that is necessary for optimality.
• 😒 The Interior Point algorithm modifies the LP problem to have an interior point and uses barrier functions to keep the constraints away from their boundaries.
• 🗽 The number of iterations in the Interior Point algorithm is at least L, where L is the bit complexity of the LP problem.
• 🔇 The volume of the ellipsoid in the Ellipsoid algorithm decreases exponentially as the number of iterations increases.

Questions & Answers

Q: What is the difference between the Simplex and Interior Point algorithms?

The Simplex algorithm moves from vertex to vertex in a polytope, while the Interior Point algorithm stays inside the polytope and gradually moves towards the optimal vertex.

Q: How does the Ellipsoid algorithm work?

The Ellipsoid algorithm uses an ellipsoid to check for the feasibility of a polytope. It starts with an ellipsoid that contains the polytope, and if the center of the ellipsoid is not in the polytope, it finds a violated constraint and creates a new ellipsoid that is guaranteed to contain only the other side of that constraint.

Q: What is complementary slackness?

Complementary slackness is a condition in linear programming where either the primal variable is zero or the dual slack is zero. It is a necessary condition for optimality.

Q: How does the Interior Point algorithm handle infeasible problems?

The Interior Point algorithm modifies the LP problem so that it has an easily found interior point. For an infeasible problem, the algorithm will detect it and output that the problem is infeasible.

Summary & Key Takeaways

• The Simplex algorithm is a polynomial time algorithm for solving linear programming problems, but its implementation takes exponential time. It finds the optimal solution by moving from vertex to vertex in a polytope.

• The Ellipsoid algorithm is another polynomial time algorithm for linear programming, which can solve exponentially sized linear programs. It works by checking the feasibility of a polytope, based on a given ellipsoid.

• The Interior Point algorithm is a polynomial time algorithm for linear programming that attempts to stay inside the polytope and gradually move towards the optimal solution. It uses barrier functions to keep the constraints away from their boundaries.