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Results tagged with linear-programming
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user 9022
Linear programming is the study of optimizing a linear function over a set of linear inequalities. The Simplex Method, Ellipsoid Method and Interior Point Method are popular algorithms to solve linear programs.
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Accepted
Large scale least squares of non symmetric and non square problems
A widely used iterative method for large scale linear least squares problems that allows for regularization if you want/need it is the LSQR algorithm of Paige and Saunders. See
http://www.stanford.ed …
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votes
Sensitivity analysis in conic optimization
In general, your problem can become infeasible under arbitrarily small perturbations $\Delta b$. For example, consider the problem
$\min X_{11}+X_{22}$
subject to
$X_{11}=0$
$X_{22}=0$
$X \suc …
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generalization from linear programming solution
You should look into the topic of "parameteric linear programming". In certain situations where the problem data vary linearly with parameters, you can derive formulas for the optimal solution and op …
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6
votes
Why are optimization problems often called "programs"?
When the term "linear programming" first came into use, computers were still very rare beasts, and the term "computer programming" wasn't that widely used. Here "programming" meant planning. As rese …
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Rewrite optimization objective
Here's a solution for the specific case where $f(x)=\| x \|_{1}$ and $g(x)=\| y-Ax \|_{2}$. The same appraoch applies to $g(x)=\| A^{T}(y-Ax) \|_{\infty}$.
There are two cases.
If $g(0) \leq s$, …
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Rewrite optimization objective
You need to start by by being explicit about the "equivalence" between the problems. Do you mean that "For every $s$, there is a $t$ such that if $x^{ * }$ is an optimal solution to the first problem …
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positive semidefiniteness: a psd matrix substracted by another rank 1 psd matrix
This is easy to formulate as a semidefinite programming problem.
First, let $X=xx^{T}$. The semidefiniteness constraint becomes
$A-\lambda X \succeq 0$
Next, use a standard technique to handle t …
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Accepted
What does "Vertex Solution" mean?
In the context of linear programming, and assuming that you're using the simplex method to solve your LP's rather than an interior point method, it's most likely that the author means "basic feasible …
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Accepted
Nonstandard Hessian approximations in Gauss-Newton
You could certainly try this, but you'd have to do a lot of careful analysis to derive any convergence results. Among other things to consider:
Your objective E(x) is more likely to have local mini …
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2
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Levenberg-Marquadt near the minima for non-zero-residual problems
The technique of multiplying the diagonal by $(1+\lambda)$ fails when you've got a 0 on the diagonal of $J^{T}J$. Having a zero on the diagonal of $J^{T}J$ can happen when you're far away from the op …
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3
votes
Accepted
Cascading minimization problems
You can solve the first LP, obtain the optimal value, and then add a constraint that the first objective has that optimal value. Then you can add your second set of constraints to the original set of …
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3
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Accepted
linear programming with OR restrictions
Your additional constraints make the feasible region for your problem nonconvex, and thus it cannot be represented as a linear programming problem.
If you can obtain an upper bound $M_{i}$ on the max …
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SDP Algorithms/ maximally complementary solutions
The standard approach is to embed the original SDP in a self dual formulation that has strictly feasible primal/dual solutions, solve the self dual formulation and then reach conclusions about the ori …
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Efficient Algorithm For Projection Onto A Convex Set
You haven't told us anything about the size of your problem instances. How many terms are there in the sum of norms? What is $n$?
Your problem is an example of a "sum of norms" optimization probl …
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Continuity of Lexicographic Minimum Solution of a parametrized LP problem
You've chosen a somewhat limited form of "parametric LP" here. It's more common to allow $F$ to vary as $F=F_{0}+tF_{1}$. For that more general variety of parametric LP, the conjecture is false- it' …
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