Hello fellow mathematicians,
I am a computer engineering PhD student specializing in control engineering, seeking your insights on an optimization method I've been working on. My approach revolves around level curves of the cost function that happen to be circumferences. Here's the crux of my inquiry:
Circumference Level Curves: In my method, the level curves of the cost function are uniquely formed as circumferences. This characteristic brings about certain geometric considerations that I believe impact the optimization process.
Unconstrained Optimal Solution and Affine Constraints: My optimization involves an unconstrained optimal solution, which I represent as a point. Additionally, there are affine constraints that collectively shape a convex polytope. I have observed that the constrained optimal solution can be found by locating the point that presents the least Euclidean distance between the polytope and the unconstrained solution. This optimal solution is derived from the point where the smallest circumference touches the polytope.
Projecting onto the Polytope Face or Vertex: When I project the unconstrained solution onto the closest face of the polytope, it seems to yield the optimal constrained solution. However, I'm encountering challenges when the unconstrained solution lies within a region that lacks a normal face for projection. In such cases, it appears that the constrained optimal solution lies at the closest vertex.
Alternative Approach with Constraint Lines: I'm contemplating an alternative approach to finding the constrained optimal solution. Instead of directly seeking the closest vertex, which might be computationally complex, I'm considering extending the lines of the constraints and projecting the unconstrained optimal solution onto the closest line. Once I have this projected point, I then saturate it to find the solution. My question is whether this approach would lead to an optimal solution, given the specific characteristics of my problem.
In essence, my query is twofold:
-Is my strategy of projecting onto constraint lines and saturating the point a viable alternative to finding the closest vertex in scenarios where normal faces are absent?
-Are there any theoretical or practical considerations I should be aware of when employing this approach, especially considering the unique characteristics of the level curves and affine constraints?
It's important to note that I only require the first optimal coordinate of the solution, which adds a specific dimension to my inquiry.
Thank you for taking the time to read and consider my question. Warm regards, Lucian.