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I am working on optimization of an objective function which is a ratio of two linear functions given as mx + b/-mx+c. Can somebody comment about convexity of this function and/or give me some reference?

sukhalid
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You might consider consulting Stephen Boyd's convex optimization book. Section 3.4 (example 3.32) is what you are interested in. Your example is called a linear fractional function and is indeed quasiconvex and quasiconcave if you restrict the domain of the denominator to be either greater or less than 0. Quasiconvex optimization problems can be solved using a method like bisection which involves solving a series of feasibility problems

Casey
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The easiest litmus test for convexivity of a function is to take the derivative and consider the region where this derivative is zero - these are potential local minima, though they could be global minima or saddle points.

In this case, your derivative is: (d)/(dx)((m x + b)/(-m x + c)) = (m (b + c))/(c - m x)^2

There is no zero point at all that depends on x, except at infinity. There is no minimum.

Andrew Charneski
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  • Dear Charneski, thanks for the answer, based on this result can we conclude that the function is non-convex? – sukhalid Jan 02 '18 at 08:36
  • Actually, consulting the strict definition of convex, this function could be convex depending on the parameters, if the derivative is always positive. You know something __is not__ convex if you can see it has local minima. My mistake. – Andrew Charneski Jan 02 '18 at 18:17