Optimization Problem: x_1(sin(x_1)) such that exp(x_1)-1>=0

[ver_mathstats]

Homework Statement

Determine if the problem is a convex optimization problem

Relevant Equations

x_1(sin(x_1)) such that exp(x_1)-1>=0

I know to solve this problem we need to see if x₁sinx₁ is convex and if the constraint is convex. I already know that x₁sinx₁ is not convex so the problem is not convex, but for proving that this function is not convex is where I am confused. But how do I go about showing this? I'm assuming I cannot use a hessian or the definition because both use two variables? So do I just find the second derivative and see if it is positive? Is that sufficient enough?

 

[Mark44]

Homework Statement:: Determine if the problem is a convex optimization problem
Relevant Equations:: x_1(sin(x_1)) such that exp(x_1)-1>=0

I know to solve this problem we need to see if x₁sinx₁ is convex and if the constraint is convex. I already know that x₁sinx₁ is not convex so the problem is not convex, but for proving that this function is not convex is where I am confused. But how do I go about showing this? I'm assuming I cannot use a hessian or the definition because both use two variables? So do I just find the second derivative and see if it is positive? Is that sufficient enough?

I took a class in linear programming many years ago, but I don't recall that it touched on convex optimization. From this link, https://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf, I dug up the basics of convex optimization.

Apparently, what you want to do is to minimize $x\sin(x)$ subject to the constraint $e^x \ge 1$. I've omitted the subscripts on x here since they serve no useful purpose in your problem. Section 1.3 of the book I linked to describes convex optimization and defines what it means for a constraint to be convex.

 

[ver_mathstats]

I took a class in linear programming many years ago, but I don't recall that it touched on convex optimization. From this link, https://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf, I dug up the basics of convex optimization.

Apparently, what you want to do is to minimize $x\sin(x)$ subject to the constraint $e^x \ge 1$. I've omitted the subscripts on x here since they serve no useful purpose in your problem. Section 1.3 of the book I linked to describes convex optimization and defines what it means for a constraint to be convex.

Thank you, I wasn't provided a textbook for this class so this is very helpful