Do separable cubic constraint and separable quartic constraint SOCP presentable?

Question

I am an engineer who is doing some network modeling and optimization. During my work, I was running into a case that is quite strange. The problem that I am trying to solve seems to be convex and it can be readily solved by any general convex solver:

\begin{array}{*{20}{c}} {\mathop {\min }\limits_{x,y,z,t,y} }&{x + y + z + t + y}\\ {}&{{a_1}{x^3} + {a_2}{y^3} + {a_3}{z^3} - t \le 0}\\ {}&{{b_1}{x^4} + {b_2}{y^4} + {b_3}{z^4} - y \le 0}\\ {}&{x + y + z + t + y \ge 1}\\ {}&{...{\rm{and}}\,{\rm{some}}\,{\rm{linear}}\,{\rm{constraints}}} \end{array}

Here, all $a_1,a_2,a_3$ and $b_1,b_2,b_3$ are positive numbers and all decision variables are non negative since they are some physical quantities.

However, in practice I see that SOCP (or QP) problem can be solve very fast by exploiting their structure. Out of pure curiosity, my question is:

Do the following separable cubic constraint and separable quartic constraint SOCP presentable ?

${{a_1}{x^3} + {a_2}{y^3} + {a_3}{z^3} - t \le 0}$

${{b_1}{x^4} + {b_2}{y^4} + {b_3}{z^4} - y \le 0}$

My initial guess is that, it is very likely since Mosek has mentioned something about power cone and exponential cone. However, I dont know how to actually do this.

An interesting example is here:

Proving the set $\left\lbrace \frac{(x + y)^2}{\sqrt{y}} \leq x - y + 5, y > 0 \right\rbrace$ is convex

Would you kindly help me with this ?

Thank you for your enthusiasm !

Answer 1

In general, for every rational number $r=p/q>1$ the inequality $t\geq x^r$ is convex on the set where $x\geq 0$ and it can be modeled using second-order cones. The model depends on $p,q$ and gets bigger as $p,q$ grow, so at some point it becomes easier to write this constraint using a more general power cone as in https://docs.mosek.com/modeling-cookbook/powo.html#powers . The power-cone representation works uniformly for all real $r>1$.

However for small values of $p,q$, and especially $q=1$ as in this case, the representation with second-order cones is reasonably nice. For example $t\geq x^4$ is equivalent to $$t\geq y^2,\ y\geq x^2$$ where both inequalities describe a rotated quadratic cone (https://docs.mosek.com/modeling-cookbook/cqo.html#rotated-quadratic-cones). Similarly $t\geq x^3$ is equivalent to $$tx\geq y^2,\ y\geq x^2$$ where again we have two rotated quadratic cones.

The general model of $x$ to an integer power follows by induction: $$t\geq x^{2n} \iff t\geq y^2,\ y\geq x^n,$$ $$t\geq x^{2n-1} \iff tx\geq y^2,\ y\geq x^n.$$