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Is function formula convex in x and y jointly? I want is to estimate both parameter x and y, that minimizes the least square. If the function is convex in both x and y jointly, then technically I can find x and y by iterating between 2 steps: Find best x given y and find best y given x.

Obviously I know I might be wrong in multiple levels. Function look non-convex as there a multiple saddle point ie. all x=0 and y=0. But if I have a constrain that y>0, this problem is no longer there. Further, I am not sure whether the iterative algorithm work and converge even if the function is convex.

vinash85
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2 Answers2

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You can compute Hessian and check whether it is positive definite.

Michael
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A convex optimization problem is defined to have convex objective, convex inequality constraints, and affine equality constraints. As you've pointed out, your objective is not convex therefore is not a convex optimization problem. This problem also seems underspecified. Why not just solve the problem minimize sum_i (a_i - alpha* b_i)^2 over alpha? This problem is convex in alpha of course and once you've found alpha, you can go ahead and choose any x and y such that x*y = alpha, though I admit it's not clear why you'd want to do this

Casey
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