# is it possible to optimally solve the following optimization problem?

I have following optimization problem $$\underbrace{\max}_{x,y,u,v,\eta} \eta \\ \text{s.t: } u\log(1+ax)\geq \eta \\ v\log(1+by)\geq\eta\\ u+v\leq1\\ux+vy\leq c\\x\geq0\\y\geq 0$$ where $a,b,c$ are some positive constants. Any help in this regard will be much appreciated. Thanks in advance.

• what about $a$ and $b$? No constraints? – sredni vashtar Apr 6 '18 at 5:34
• @Sun edited. $a,b$ are constants – Frank Moses Apr 6 '18 at 5:40
• Is $ux+uy\le c$ perhaps a typo for $ux+vy\le c$? I just ask because it breaks the $u-v$ symmetry I see in the other inequalities. – saulspatz Apr 6 '18 at 5:49
• @saulspatz spot on. It was a typo. Corrected now. – Frank Moses Apr 6 '18 at 6:13
• It seems plausible that the optimum occurs when all the constraints (except $x,y\ge0$) are active, i.e. $u\log(1+ax)=\eta=v\log(1+by)$, $u+v=1$, $ux+vy=c$. Have you tried writing out the KKT conditions? – user856 Apr 6 '18 at 6:59

Depends on what you mean with possible? Of course you can just throw the problem at any nonlinear solver. Here I use the MATLAB toolbox YALMIP (disclaimer, developed by me) and use its internal spatial branch&bound global solver to compute a globally optimal solution. The global solver requires bounds on all variables, so I just added some without much thought, I think you can derive valid upper bounds on $x$ and $y$ easily from the bounds you have.
a = 1; b = 2; c = 3;