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

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In summary: So basically what I need to do is to find the second derivative of ##x\sin(x)## and see if it is positive?
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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?
 
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ver_mathstats said:
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.
 
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Mark44 said:
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
 

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

What is an optimization problem?

An optimization problem is a mathematical problem in which the goal is to find the best possible solution from a set of possible solutions. This is typically done by maximizing or minimizing a specific objective function while satisfying a set of constraints.

What is the objective function in this optimization problem?

The objective function in this optimization problem is x1sin(x1).

What are the constraints in this optimization problem?

The constraint in this optimization problem is that exp(x1) - 1 must be greater than or equal to 0. This ensures that the solution is within the domain of the objective function.

What is the significance of the objective function in this optimization problem?

The objective function represents the quantity that is being optimized. In this case, it is the product of x1 and sin(x1). By finding the optimal value of x1, we can maximize the value of the objective function.

How can this optimization problem be solved?

This optimization problem can be solved using various mathematical techniques such as calculus, linear programming, or heuristic algorithms. The specific method used will depend on the complexity of the problem and the available resources.

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