两个线性函数之比的凸性咨询：mx+b/(-mx+c)目标函数优化

Hey there! Let's break down the convexity of your ratio-of-linear-functions objective step by step, focusing on single-variable convexity rules since this is a univariate function.

Key Background: Convexity for Univariate Functions #

For a univariate function ( f(x) ) defined on an interval, it's convex if its second derivative ( f''(x) \geq 0 ) for all ( x ) in the interval; concave if ( f''(x) \leq 0 ). If the second derivative changes sign across the domain, the function isn't convex (or concave) over its entire domain.

Step 1: Define the Function and Its Domain #

Let ( f(x) = \frac{mx + b}{-mx + c} ). First, note the domain excludes ( x = \frac{c}{m} ) (when ( m \neq 0 )) since the denominator can't be zero. If ( m = 0 ), the function simplifies to ( \frac{b}{c} ) (a constant), which is both convex and concave (trivially, as it satisfies the convexity inequality with equality everywhere).

Step 2: Compute Derivatives (for ( m \neq 0 )) #

Let's calculate the first and second derivatives to analyze convexity:

• First derivative:
  [
  f'(x) = \frac{m(-mx + c) - (mx + b)(-m)}{(-mx + c)^2} = \frac{m(b + c)}{(-mx + c)^2}
  ]
• Second derivative:
  [
  f''(x) = \frac{-2m^2(b + c)}{(-mx + c)^3}
  ]

Since ( m^2 > 0 ) (for ( m \neq 0 )), the sign of ( f''(x) ) depends on two factors: ( -(b + c) ) and ( (-mx + c)^3 ).

Step 3: Analyze Convexity by Interval #

The domain splits into two disjoint intervals: ( x < \frac{c}{m} ) and ( x > \frac{c}{m} ):

• Case 1: ( b + c = 0 )
  The second derivative becomes 0 everywhere in the domain, so ( f(x) = -1 ) (a constant). Again, this is both convex and concave.

• Case 2: ( b + c \neq 0 )

  • For ( x < \frac{c}{m} ): ( (-mx + c)^3 > 0 ) (since ( -mx + c > 0 )). Thus:
    • If ( b + c > 0 ), ( f''(x) < 0 ): the function is concave on this interval.
    • If ( b + c < 0 ), ( f''(x) > 0 ): the function is convex on this interval.
  • For ( x > \frac{c}{m} ): ( (-mx + c)^3 < 0 ) (since ( -mx + c < 0 )). Thus:
    • If ( b + c > 0 ), ( f''(x) > 0 ): the function is convex on this interval.
    • If ( b + c < 0 ), ( f''(x) < 0 ): the function is concave on this interval.

Summary #

Your objective function is not convex over its entire domain (unless it's a constant function). Its convexity switches between the two intervals separated by ( x = \frac{c}{m} ).

Reference Materials #

For deeper context on convexity of fractional functions, check out Convex Optimization by Boyd and Vandenberghe. The book covers linear fractional functions (a multivariate generalization of your function) and explains how their convexity/concavity behaves over their respective domains.

内容的提问来源于stack exchange，提问作者sukhalid