Gradient ascent with constraints

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SUMMARY

The discussion centers on optimizing a convex function F(x,y) using gradient ascent while adhering to constraints on the variables x and y. The participants emphasize the need to incorporate these constraints directly into the gradient ascent algorithm, particularly when the maximization path encounters an upper bound on x, denoted as U[x]. Instead of relying on Lagrange multipliers, the approach involves adjusting the optimization path to maintain compliance with the constraints, allowing for conditional updates to y while keeping x fixed at U[x] when necessary.

PREREQUISITES
  • Understanding of convex functions and their properties
  • Familiarity with gradient ascent optimization techniques
  • Knowledge of constraints in optimization problems
  • Basic concepts of steepest descent methods
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  • Study the implementation of gradient ascent with constraints in Python using libraries like NumPy
  • Learn about projection methods for constrained optimization
  • Explore the relationship between convex functions and Lagrange multipliers
  • Investigate the use of penalty methods in optimization to handle constraints
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Mathematicians, data scientists, and machine learning practitioners who are involved in optimizing functions under constraints, particularly those utilizing gradient ascent methods.

eren
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Hi,
I have a convex function F(x,y) that I want to optimize. Since, derivative of F does not closed form, I want to use gradient ascent. The problem is, I have constrains on x and y.
I don't know how to incorporate this into gradient search. If there was a closed form, I would use Lagrange multipliers. What should I do in this case?
Thanks,
 
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You should find a way to incorporate the constraints into your programming of steepest ascent, such that if the maximization path meets a constraint (say an upper bound on x, U[x])) then it should set x = U[x] and follow the steepest descent conditional on x = U[x] (that is, by changing y only). It should continue doing so until and unless x becomes free again (x < U[x]).
 
Thanks a lot for the reply.
Does this mean it has nothing to do with Lagrange multipliers? Indeed, I have a concave function F(x,y) in which x and y are vectors and the constraints are upper bounds on the magnitudes of x and y. What exactly do you mean with "the maximization path meets the constraint" ? The steepest descent from my understanding does not usually meet the constraint.
 

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