# Tag Info

### What is required of the objective function in order to use Gauss Newton method?

Just to clarify notation, I'll be discussing the Gauss-Newton method for the problem $\min \phi(x)=(1/2) \| F(x) \|_{2}^{2}$ with the search direction $p$ computed as the solution to the linear ...
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### Looking for name of optimization problem in form $\min \mathrm x^T \mathrm A \mathrm x$ subject to $\|\mathrm x\| = 1$

Since the optimization problem has a well-known closed-form solution, it is rarely used in itself and hence usually not given a name. The objective function $x^TAx$ (using $\|x\|=1$), however, is ...
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### How to determine whether two cylinders intersect or not?

David Eberly has a good description of the solution (with pseudocode) here: http://www.geometrictools.com/Documentation/IntersectionOfCylinders.pdf The short summary: like most convex-convex ...
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### Linear system solution with inequality constraints - methods?

This is a linear programming feasibility problem (since you don't have an objective function to minimize or maximize.) You can simply use an objective function of $0$ and hand this off to any ...
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### Integration of differential equation with orthogonality constraint

In general, one cannot expect rk4 to maintain quadratic invariants of the system, it simply doesn't do that. Methods that do maintain specific invariants have to be specially devised — this usually ...
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### Intersections of supports constraint

Suppose that $\mathbf{x}\ge0$ and $\mathbf{y}\ge0$. Then a necessary and sufficient condition for $\mathrm{supp}\{\mathbf{x}\}\cap\mathrm{supp}\{\mathbf{y}\}=\emptyset$ is for the two vectors to be ...
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### How to create an optimal pizza delivery plan and how to visualize it

Problem Formulation I can't guarantee that this is a perfect (or smallest-possible) formulation of the problem, but maybe it will help guide a better one. The road network is a directed graph ...
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### Scaling/nondimensionalization for numerical optimization

One thing that makes your non-dimensionalized ODE confusing is that you use the same symbols for dimensionalized and nondimensionalized variables, even though they are different variables. Consider ...
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Let's consider the following minimization problem \begin{align} & \text{minimize} && \tfrac12 z^\mathrm{H} P_0 z + q_0^\mathrm{H} z\\ & \text{subject to} && \tfrac12 z^\mathrm{...
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### Optimization with the constraint of rank=1

You can parameterize your rank-1 matrix $A$ as $A=xs^{T}$ where $x$ and $s$ are the unknown $n$ by $1$ column vectors. You then have an unconstrained problem $\min f(x,s)$. Depending on the ...
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### Looking for open source numerical solver

IPOPT is a good interior point method solver for convex nonlinear problems, and has a MATLAB interface, although I haven't used the MATLAB interface. (The solver, called from GAMS, is very good.) CVX ...
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### Find $\min x^TAy$ subject to $1^Tx=1^Ty=1,x\ge 0,y\ge 0$

If there's no mistake then it's much easier than I thought. We will show that the minimum is equal to the smallest component of $A$, denoted by $a_{i_0j_0}$ where $(i_0,j_0) = \arg\min_{(i,j)} a_{ij}$...
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### Converting convex quadratic constraint to linear matrix inequality (LMI)

One option here is to use the pseudoinverse of $\Sigma$ rather than the actual inverse. Appendix A of Boyd and Vandenberghe discusses a version of the Schur complement that includes this case. ...
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### Rank constrained SDP

The constraint $\mbox{rank}(X) <= d$ is in general a non-convex constraint. Sorry. A commonly used approach is to minimize the Schatten 1-norm of X (the sum of singular values of X) as a ...
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### How to understand what's going wrong in a code for solving a problem with augmented lagrangian?

It's very difficult to provide a useful answer to this question because you haven't provided enough details about your problem and what you have already tried. There are various conditions under ...
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### How to debug a constrained optimization algorithm?

You can use a Test Driven Development (TDD), the ideas behind are: before write code you write the test with the expected result; every line of code (almost all lines) is under test so when you ...
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### I have to solve a large binary programming task. Should I avoid branch and bound?

10,000 variables is a lot for an integer programming problem, but everything depends on the details of your particular problem. With the information provided, there's really no way for us to tell you ...
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### How to test convergence of an algorithm for constrained optimization

The first order necessary optimality conditions for a minimization problem with inequality constraints are not that the gradient vanishes, since the minimum can be attained at the boundary of the ...
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