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.

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How to minimize $\| A x - b \|_1$

How can I minimize $\|A x - b\|_1$ where $A$ is not invertible? Here, $x\in\mathbb{R}^N$. I know that I can use the sub-gradient method, but that would be very slow. I also know that I can use the ...
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What is the largest ellipse of given eccentricity that can be inscribed into square?

If eccentricity of ellipse is known, what is the position of the biggest such ellipse that can be inscribed into square? I could find the answer on the web – the biggest ellipse is positioned with its ...
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Proposition 3.2. of Bertsekas' paper about Lagrange multipliers

This is a problem about the proposition 3.2 of Bertsekas' paper below. Bertsekas D P, Ozdaglar A E. Pseudonormality and a Lagrange multiplier theory for constrained optimization[J]. Journal of ...
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I have a question regarding the McCormick envelope in mathematical optimization. The McCormick envelope allows to create a convex relaxation for a bilinear term. Given the bilinear constraint $w = x \... • 53 0 votes 0 answers 17 views Berge's Maximum Theorem for Parameters that are Functions I am given some functions$f_i$that are$C^1$,$f_i : \mathbb{R}_+ \rightarrow \mathbb{R}_+$and some budget$z \in \mathbb{R}_+$. I have some constrained optimization problem with a continuous ... • 23 0 votes 0 answers 22 views Minimize function two variables II. Let$x,y,a,b$be the column vectors$(n,1)$,$C(n,n)$be the matrix, and $$\phi(x,y)=\left\Vert x-a\right\Vert _{2}^{2}+\left\Vert y-b\right\Vert _{2}^{2}+\left\Vert x^{T}Cy\right\Vert _{2}^{2}$$ ... • 383 0 votes 0 answers 26 views Is it possible to convexify the inequality constraint$z \leq x^3 \cdot y$? Is there a way to convexify the inequality constraint$z \leq x^3 \cdot y$in a nonlinear optimization problem with$x, y, z$being nonnegative variables? • 53 1 vote 0 answers 22 views How to formulate piecewise quadratic function optimization without introducing binary variables? I have a problem with logical constraints (either-or constraints). I know that it can be solved by either big-M or complementary formulations. However, i do not want to convert it into mixed-integer ... 0 votes 0 answers 55 views Minimize function of two variables Let$x,y$be the column vectors$(n,1)$,$A(n,n)$be the matrix, and $$\Phi(x,y)=\left\Vert x^{T}Ay-y^{T}Ax\right\Vert _{2}^{2},$$ be the minimized function of two variables$x,y$. The function can be ... • 383 1 vote 2 answers 61 views Projectile Motion Optimization - Minimum Final Velocity? I'm doing a project on the physics of basketball and wanted to find the optimal angle of release for a free throw (the basket is 3.05 m tall and 4.572 m away). The optimal angle would be the angle at ... • 207 2 votes 1 answer 50 views Shortest path on the surface of a cylinder between given points$A$and$B$Suppose you have the cylinder$ x^2 + y^2 = R^2 $And points$A = (R, 0, 0)$and$ B = (0, R, h) $. Find the parametric equation of the curve of shortest length connecting$A$and$B$. My attempt: If ... • 23.5k 0 votes 1 answer 37 views Matrix subset selection We aim to select rows and columns of any matrix$\mathbf{A}\in\mathbb{R}^{m\times n}$. Define a selection matrix$\mathbf{S}\in\mathbb{R}^{m\times n}$where$S(i,j)=x_i \cdot y_j$, the matrix after ... • 75 0 votes 1 answer 54 views Optimizing an Objective Function While Minimizing an Argument? I'd like to preface this by noting I'm not too familiar with optimization techniques, so something may or may not be off. Suppose I have some scalar function$f\left(x_1, x_2\right)$, where$x_1,x_2\...
I want to minimize the following function (with $<A,B> = tr(A^T B)$) $<\Theta, \hat \Sigma> - \log det \Theta + \lambda_n \lVert \Theta \rVert_{1, off}$ over the set of symmetric positive ...