# Formulate and solve a simple conic programs in cvxpy language [closed]

Let $$r,\epsilon > 0$$ and $$a, b \in \mathbb R^n$$ with $$\|a\|_2 \le r$$. Define $$C(a) := \{x \in \mathbb R^p | \|x+a\|_2 \le r,\;\|x\|_\infty \le \epsilon\}$$, and assume it is non-empty.

# Question

• (A) How to formulate and solve the problem problem $$\sup_{x \in C(a)} b^Tx$$ in the cvxpy language ?
• (B) Same question with $$\|a\|_2 = r$$ and $$C(a) := \{x \in \mathbb R^p | \|x+a\|_2 = r,\;\|x\|_\infty \le \epsilon\}$$

Disclaimer: I've never done cvxopt / cvxpy before. I plan to learn the syntax later. For now, I just want something to plug-and-play. Thanks!

## 1 Answer

You can solve Question A as a second-order cone program like so:

#!/usr/bin/env python3

import cvxpy as cp
import numpy as np

##########################
#Question A
##########################

n = 50                          #Arbitrary number of dimensions
r = 10                          #Arbitrary radius
e = 3                           #Epsilon value
a = np.random.random(size=50)   #Generate random a vector
a = a/np.linalg.norm(a, ord=2)  #Scale to unit length
a = r*a                         #Scale to radius

x = cp.Variable(shape=n)        #x, a variable to be optimized
cons = []                       #List of constraints

#See cvxpy's atoms here: https://www.cvxpy.org/api_reference/cvxpy.atoms.html
cons += [cp.norm(x+a,2)<=r]     #Note that we are using cvxpy's norm function!
cons += [cp.norm_inf(x)<=e]

#Objective
obj = cp.Maximize(cp.sum(a*x))

#Formulate problem
prob = cp.Problem(obj, cons)

#Solve problem
optval = prob.solve()

print("Optimum value = {0}".format(optval))
print("x = {0}".format(x.value))


Question B you can't solve using plug-and-play with cvxpy because the problem is non-convex (you're optimizing over the surface of an ellipsoid).

• Thanks! BTW, i fixed a typo in the question: objective should be $b^Tx$ not $a^Tx$. – dohmatob Jan 22 '19 at 2:05
• @dohmatob: Were you able to figure out how to incorporate that typo fix into the code above? – Richard Jan 27 '19 at 15:22
• lol, sure. BTW, your answer above was precisely what i needed to get me familiarized with syntax and semantics of cvxpy / cvxopt. Thanks! – dohmatob Jan 27 '19 at 15:42
• @dohmatob: Happy to be of service. If this answer adequately answered your question feel free to upvote it by clicking the gray arrow next to it, and/or to accept it by clicking the outline of the checkmark :-) Also, cvxpy has discplined geometric programming which can solve a large class of interesting problems using a similar syntax. – Richard Jan 27 '19 at 16:03
• Answer accepted. Thanks for the ref on DGP. – dohmatob Jan 28 '19 at 7:59