询问给定优化问题能否重构为geometric program及是否存在convex reformulation

Great question! Let's break this down step by step.

Geometric Program (GP) Reformulation #

First, recall that a geometric program requires:

• The objective function to be a posynomial (sum of monomials with positive coefficients, where variables are strictly positive)
• Constraints to be posynomial inequalities or monomial equalities

Looking at your objective function ( P(x,y,z) = \frac{x}{2x+3y} + \frac{y}{y+z} + \frac{z}{z+x} ):
Each term is a ratio of a monomial to a posynomial (sum of monomials). Posynomials are closed under addition, multiplication, and positive scaling—but not under division by posynomials. So each term in ( P ) is not a posynomial, and their sum isn't either.

Is there a substitution that could turn this into a GP? Unfortunately, no common transformation fixes the fractional structure here:

• Reciprocating variables just permutes the terms without resolving the additive denominators
• Logarithms or other algebraic transformations can't convert the sum of fractional terms into the multiplicative structure required for posynomials

So this problem cannot be reformulated as a geometric program—its objective doesn't fit the required form of GP objectives or constraints.

Convex Reformulation #

Next, let's address convexity. First, is ( P(x,y,z) ) itself convex over the feasible region? Let's test a concrete example:
Take two feasible points: ( (4,1,1) ) where ( P ≈ 1.063 ), and ( (4,4,4) ) where ( P = 1.2 ). The convex combination ( (4, 2.5, 2.5) ) gives ( P ≈1.142 ), which is greater than the convex combination of the two ( P ) values (( 0.5*(1.063+1.2)=1.1315 )). This violates the convexity definition, so ( P ) is not convex.

Can we find an exact convex reformulation using substitutions or auxiliary variables?

• Each fractional term is quasiconvex/quasiconcave, but their sum is not guaranteed to be either
• Techniques like the Charnes-Cooper transformation work for single fractional objectives, but fail for sums of fractions—resulting in non-convex bilinear constraints
• Upper-bounding each term with auxiliary variables leads to constraints like ( t_1(2x+3y) ≤x ), which are bilinear and non-convex

That said, you can still solve this problem effectively:

• The feasible region is compact (variables bounded between 1 and 4 with ordering constraints), so gradient-based numerical methods (like interior-point methods) will find the global minimum
• As a university entrance exam problem, there's likely an analytical solution—try testing boundary points (e.g., ( x=4, y=1, z=1 )) or symmetric cases to identify the minimum.

备注：内容来源于stack exchange，提问作者Tuong Nguyen Minh