Convex Relaxations in Power System Optimization: Convex Relaxation (6 of 8)

Summary

This video introduces convex relaxations for non-linear optimization problems, focusing on power system optimization. It explains the concept of a feasible space, where variables satisfy all constraints. The video contrasts convex and non-convex feasibility spaces using examples, including power flow equations. It discusses the importance of relaxations, which extend the feasible space to include all original points, ensuring the true optimal solution isn't missed. The video also covers the benefits of convexity for optimization, such as guaranteed global optima and fast convergence, and the use of relaxations to prove infeasibility and improve solutions by tightening or adding constraints.

Takeaways

• 📐 **Feasibility Space**: The set of all possible solutions that satisfy the constraints of an optimization problem.
• 🔑 **Convex Feasibility Space**: A feasibility space where any two points can be connected by a line segment that remains within the space.
• 🌐 **Non-Convex Power Flow Equations**: Power flow equations often result in complex, non-convex feasibility spaces with disconnected sections.
• 🔄 **Relaxation**: An extension of the feasible space that includes all feasible points but may also include infeasible points.
• 🔵 **Convex Relaxation**: A type of relaxation that is convex and can provide a guaranteed lower bound for minimization problems.
• 🔒 **Nested Relaxations**: Some relaxations can contain other relaxations, allowing for a hierarchy of increasingly tighter approximations.
• 🚫 **Not a Relaxation**: An approximation that does not necessarily contain all feasible points and may exclude the true optimal solution.
• 🌟 **Convexity Importance**: Convexity ensures that optimization algorithms converge to the global optimum and are numerically stable.
• 📉 **Bounding Objective Values**: Relaxations provide a lower (or upper) bound on the objective function value of the original problem.
• ❓ **Optimality Gap**: The difference between the solution of the relaxation and the feasible solution of the original problem, indicating the quality of the relaxation.

Q & A

What is a feasibility space in optimization problems?

A feasibility space represents the set of variable assignments that satisfy all the constraints of an optimization problem.

What happens when you add constraints to the feasibility space?

When constraints are added, the feasible space is restricted, often resulting in a smaller, more defined region that satisfies the new conditions.

What is the difference between a convex and non-convex feasibility space?

A convex feasibility space allows any two points within it to be connected by a straight line that remains within the space, while a non-convex feasibility space can have regions that are disconnected or irregular in shape.

What is a relaxation in the context of optimization problems?

A relaxation is an extension of the feasible space that contains all the feasible points of the original problem, and may include additional points that are not feasible for the original problem.

How does a relaxation help in solving optimization problems?

A relaxation provides a broader space in which to search for solutions, making the problem easier to solve, and it ensures that the true optimal solution is never cut off.

What is the significance of convex relaxations?

Convex relaxations are significant because they ensure fast and guaranteed convergence when solving the problem, making them easier to compute than non-convex problems, and they provide a lower bound for minimization or upper bound for maximization problems.

What is the difference between a relaxation and an approximation?

A relaxation is an extension of the feasible space that contains all feasible points, while an approximation is a representation of the feasible space that may not include all feasible points or might include infeasible ones.

Why are relaxations useful for proving infeasibility?

Relaxations are useful for proving infeasibility because if the relaxation has no feasible solutions, it can be used to conclude that the original problem also has no solution.

What is the convex hull and why is it important in optimization?

The convex hull is the smallest convex set that contains all the points of a non-convex set. It is important because it represents the tightest convex boundary of a problem’s feasible region, and finding it is crucial to creating effective relaxations.

How can relaxations be improved for tighter approximations?

Relaxations can be improved by intersecting different relaxations, adding extra constraints known as valid inequalities, or by using more refined methods to approximate the feasible region more closely.