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Questions tagged [optimization]

Optimization is the process of choosing the "best" value among possible values. They are often formulated as questions on the minimization/maximization of functions, with or without constraints.

In mathematics, computer science, economics, or management science, mathematical optimization (alternatively, optimization or mathematical programming) is the selection of a best element (with regard to some criteria) from some set of available alternatives.

An optimization problem can be represented in the following way: given a function $f:A\to\mathbb{R}$ from some set $A$ to the real numbers, we want to find an element $x_0\in A$ such that $f(x_0)\le f(x)$ for all $x \in A$ ("minimization") or such that $f(x_0)\ge f(x)$ for all $x \in A$ ("maximization").

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What's the meaning of extra variables present in constraints, but not in the objective?

What's the meaning of extra variables present in constraints, but not in the objective? E.g. $$\min (x_1,x_2)$$ $$s.t. x_1+x_2-x_3 \geq 1$$ Does this mean that $x_3$ is redundant, since it's not present in the objective?
mavavilj
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Show that an LP is unbounded

I am working on the following exercise: Consider the LP $max \ c \cdot x$ with $Ax \le 0$ and $x \ge 0$ Show that either $x=0$ is an optimal solution or the LP is unbounded. My idea for a proof goes as follows: We assume that $0$ is not an…
3nondatur
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Thoughts on how to solve this optimization problem using KKT?

I came across this optimization problem but I did not learn optimization theory previously. Could anyone provide some thoughts on how to solve them? $ \begin{align} \text{minimize} & \quad e^{-\frac{{t_1}^2}{2}} + e^{-\frac{{t_2}^2}{2}} +…
Alex Gao
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Need to solve a constrained optimization problem but lacks the software to do so

I need someone to compute a constrained optimization problem. I couldn't find any software that could handle an optimization problem as long as this one is, and I don't want to resort to hand calculating it just yet. Here's the problem: Maximize…
Giuseppe Luchitta
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coordinate descent in very basic

I try to figure out how coordinate descent works from wiki https://en.wikipedia.org/wiki/Coordinate_descent From wiki example : the equation is $5x^2-6xy+5y^2$. Let $x = -0.5$ and $y =-1$ For the first iteration $y$ is going up graphically and the…
Linear Algebra fans
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Sufficient condition for a point on the boundary of a feasible set to be a minimizer

Consider the optimization problem minimize $-2x^2-y^2$ subject to $x+y=2$ $x\geq0$ $y\geq0$ The global minimum of this problem is when $y=0$ and $x=2$. How can we check the second order sufficiency condition at this point? The Lagrangian…
Eric Brown
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Maximum perimeter of triangle in a rectangle

What is the maximum perimeter of a triangle that is inscribed in a rectangle? Prove your claim I found this problem while formulating the question of maximum length of rod that a truck can carry with the shape of wire being an arbitary…
MATHS MOD
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Matrices and Vectors LINGO

I am working in LINGO software. I have two matrices X and Y with different sizes. I need to write all the elements of the two matrices in one unit vector Z. Did any one encountered such a case before and was able to do it?
user321821
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Optimization: Coupling Variables

I have formulated an optimization problem that is expressed as: $$\begin{align*} \underset{\mathbf{x}}{\text{minimize}} &&-\log(ax_1)-\log(bx_2)\\ \text{s.t.} && x_1 \leq 0.2\\ \text{} && x_2\leq x_3 \\ && x_1+x_2+x_3=1 \\ \end{align*}$$ where…
M.A.N
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Uncertainty sets for robust optimization

I have a few questions regarding the Robust Optimization(RO) approach. I am trying to consider uncertainty in a rhs vector, which is uncertain demand. First, Im not seeing any different in proposing a box uncertainty set, and just putting the…
Alireza Amani
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How to optimize $ \inf_{\epsilon \geq 0} \{\epsilon + \frac{r\sqrt{2B}}{n}\}$ subject to $B \leq C\epsilon^{-1/\alpha}$?

Curious about how to solve this optimization problem, anyone has some thoughts about this? $$ \inf_{\epsilon \geq 0} \{\epsilon + \frac{r\sqrt{2B}}{n}\}$$ $$B \leq C\epsilon^{-1/\alpha}$$
rifle123
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Reformulating $\arg\min |x|$.

Is it possible to reformulate next optimization problem $\arg\min |x|$ in terms of constrained minimization problem with twice continuously differentiable functions. As an example of such transformation: $\arg\min \max_{i-1,\dots,m} (a_i^T x +…
ashim
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Global minimum of a function of 3 variables

The function is $f(x_1,x_2,x_3) = e^{x_1^2}+x_2^4+x_2^2x_3^2+x_3^4+6x_2+6x_3$ The gradient is $∇f(x)=[2x_1e^{x_1^2} ,4x_2^3+2x_2x_3^2+6, 4x_3^3+2x_3x_2^2+6]$ Now to find the extremum points the gradient must be zero and the only point I could find…
Nick202
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I need to make the output of a variable positive, for both negative and positive inputs

I need to make the output of a variable positive, for both negative and positive inputs. Since I am using it in the linear optimization problem, no squares, self-division or self-multiplication is allowed. Need a mathematical solution, No prebuilt…
Manish Arora
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formulating conflicting optimization parameters

Imagine you are trying to solve an optimization problem and your goal is to pick a set of parameters, $p_1$ to $p_i$ that a given cost function $f$ is minimized based on a set of constraints. What does it mean mathematically when two of these…
iCode
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