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I am studying a function arising in the analysis of robust aggregation rules in distributed learning, but the question is purely analytical. The function I am facing depends on parameters $a, b > 0$ and is defined as $$ f(x) = \sqrt{1 - x^2} + (ax+b)x, $$ where $x \in [0,1]$.

I want to understand the behavior of $f$, in particular, I need to understand $\max_x f(x)$.

Can the maximum be explicitly computed or at least sharply upper-bounded in closed-form?

Any thoughts or response will be appreciated.

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  • $\begingroup$ OP is supposed to ask on equestion at a time -1 $\endgroup$ Commented yesterday
  • 1
    $\begingroup$ I updated the question. $\endgroup$ Commented yesterday
  • $\begingroup$ good question .......... +1 $\endgroup$ Commented 2 hours ago

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I'll give you a road map. A complete solution seems to need some work.

I would take the derivative of $ f $, and then find the critical points, where the derivative doesn't exist or equals zero, and finally compare the values of $ f $ at the critical points, and those at the interval boundary points, to find the maximum.

1. Find critical points

Take the derivative of $ f $, which can be done manually.

1.1 Derivative doesn't exist

The derivative doesn't exist at $ x = 1 $, which is also a boundary point.

1.2 Derivative is zero

To solve $ f'(x) = 0 $, I came up with a fourth degree equation with coefficients in terms of $ a $ and $ b $: $$ 4a^2x^4 + 4abx^3 + (-4a^2 + b^2 + 1)x^2 - 4abx - b^2 = 0 $$ You can find an analytic solution for this using the root formulas for the quartic equation, or using a computer algebra system.

I would use SymPy because I am already familiar with Python. SageMath is another good option. There is of course others.

The following Python code can be run in a SageMath Cell:

from sympy import Symbol

a = Symbol('a')
b = Symbol('b')

solve(4*a**2*x**4+4*a*b*x**3+(-4*a**2+b**2+1)*x**2-4*a*b*x-b**2==0, x)

After you have an expression in terms of $ a $ and $ b $ for $x$ values that make the derivative zero, you will need to put that expression in this inequality $ 0 \leqslant x < 1 $ to find values of $ a $ and $ b $ that generate $ x $ values between $ 0 $ and $ 1 $. You can probably use a symbolic solver for this, too.

2. Compare values of $ f $ at critical points

Compare values of $ f $ at critical points that you have found above to find the largest.

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  • $\begingroup$ Indeed, this transform the problem into a quartic equation, which suggests no simple formula. I will try to push in that direction. Thanks. $\endgroup$ Commented 21 hours ago
  • $\begingroup$ nice answer ............ +1 $\endgroup$ Commented 2 hours ago

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