# Questions tagged [convex-optimization]

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### Has compressed sensing been generalized to convex optimization problems?

Has the theory of "compressed sensing" been generalized to any classes of convex optimization problems? I need to analyze a problem of the type $$\min ||x||_0, ~~~~ \mbox{ subject to } ~~~g(x) \leq 0$$...
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### SVM - running time for detecting if data is linearly separable?

If my understanding is correct, one way to check if a set of $m$ data points is linearly separable is to use support vector machines to find a maximum margin hyperlane for separating the data; the ...
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### Time complexity of standard semidefinite programming solvers

I am interested in exact scaling of the ellipsoid method and interior point methods for solving SDPs. (I am not interested in algorithms like multiplicative weights updates method.)
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### What's the state of the art for matrix nuclear/trace norm optimization

I am interested in simple matrix optimizations with nuclear/trace norm: $\min_X \left(f(X) + \|X\|_*\right)$ where $\|X\|_*$ stands for the trace norm of the matrix $X$, and $f$ is a convex smooth ...
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### Looking for an easy/pedantic exposition of Renegar's famous result on polynomial optimization

In September $1989$, Renegar had this famous sequence of 3 papers titled, "On the Computational Complexity and Geometry of the First-order Theory of the Reals, Part I/II/III". I was wondering if ...
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### Convex optimization: is it possible to find solutions that are exactly feasible and approximately optimal in polynomial time?

In Nemirovxki's lecture notes on interior point methods, I found the following. He defines an approximate solution as satisfying the following, for any given $\epsilon>0$: that is: the ...
132 views

### Which variant of the ellipsoid method was used for the Santa Claus problem?

As one of the steps in the article The Santa Claus problem (Bansal and Sviridenko, 2006) the following linear problem was considered (at the end of the second page, as the dual): \begin{align*} &\...
106 views

### Bounding a Solution of an SDP

It's common for convex optimization procedures to require a bounded region containing an optimal solution, either as input, like the initial ellipsoid of the ellipsoid method, or for run time bounds, ...
102 views

### Software that generates and solves a Lasserre hierarchy

Suppose L is a linear program that is a relaxation of some 0/1 integer linear program ILP. There is a systematic way to construct SDP relaxations of ILP that are tighter than L by using a Lasserre ...
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### Linear programming optimization problems using parallel algorithms

I'm looking for methods and algorithms for solving linear programming algorithms, characterized by up to 20 variables but up to thousands of constraints in a parallel way. There are several approaches ...
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### Has Khachiyan/Porkolob's convex integer optimization been implemented?

Khachiyan and Porkolab in 'Integer optimization on convex semialgebraic sets' gave an $O(ld^{ O(k^4)})$ algorithm to minimize a degree $d$ form with integer coefficients of binary length at most $l$ ...
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### Non-Linear Programming with \min operator in the constraint

Can the following non-linear program be solved in polynomial time? $c_{ij}$'s are constants and known. Each $c_{ij}$ is either -1 or 1. \begin{align} \text{maximize } &\sum_{i,j=1}^{m,n} c_{ij}...
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### Convergence of online convex optimization methods

I am new to this subject so this question might seem a bit trivial Assume that in each round $t\in{{1,...T}}$ we choose $x_t\in K$ where $K$ is a compact and convex set, The common methods for ...
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### Runtime of Gomory's Cutting Plane Algorithm

I read in several sources that the use of Gomory's cuts exclusively in Integer Programming was shown to be inefficient in practice when Gomory had created them. But later down the line they were shown ...
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### KKT-like conditions for values close to optimal solution

The KKT conditions are necessary and sufficient conditions for problems where we maximize over a convex function subject to linear inequality and equality constraints. That is $x^*$ is an optimal ...
1 vote
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### What is the meaning of loss in online convex optimization?

I am studying online convex optimization, and it is stated that when we make a decision, we observe loss corresponding to our decision. In some problems like multi-armed bandit problems, we know the ...
1 vote
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### Optimization as LP on the convex hull of its solution space

I have heard claims that for a certain class of optimization problems, one can re-write the problem as a linear-programming problem on the convex hull of the solution space of the original problem (...
1 vote
The subgradient algorithm for minimizing a convex function $f(x)$ is the update rule $$x(t+1) = x(t) - \alpha(t) d(t)$$ where $d(t)$ is any subgradient of $f(x)$ at $x(t)$ and $\alpha(t)$ is a ...