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Results tagged with convex-optimization
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user 75420
Optimization with convex constraints and convex objectives; notions related to convex optimization such as sub-gradients, normal cones, separating hyperplanes
1
vote
How to project a vector on a set defined by linear inequality constraints through KKT condit...
Unlike textbook examples, most optimization problems don't have closed form solutions for the KKT conditions. I believe the inequality constraint renders this such a case.
The projection you seek can …
- 1,517
0
votes
least square optimization under positive semidefinite constraint
This is easy to formulate and solve (presuming it's not too large or otherwise unpleasant) in CVX or YALMIP.
CVX:
cvx_begin
variable X(length(a),length(a)) semidefinite
minimize(norm(a*X-b))
cvx_end …
- 1,517
3
votes
Convex optimization closed-form solution
The problem is convex and is a Second Order Cone problem (SOCP) for $\beta \le 0.5$. The statement that the problem is convex for $\beta > 0.5$ is incorrect.
There is no closed-form solution.
If $\bet …
- 1,517
1
vote
Lagrange Multipliers for two constraints, degenerate case
If $\nabla g$ and $\nabla h$ are linearly independent at a given point, then the Linear Independence Constraint Qualification (LICQ) is satisfied, and presuming that $f$, $g$, and $h$ are all continuo …
- 1,517
-1
votes
maximization of a log norm function
Adding to the answer by @Robert Israel (which I'm sure he knows, but didn't mention):
Because log is strictly monotonically increasing, this problem is equivalent to $maximize \left( \|x\|_\infty \r …
- 1,517
1
vote
Has the following generalization of monotropic programming been studied in the literature?
The Extended Monotropic Programming problem deals with a more general extension to all n variables at a time than the two variables at a time extension you are interested in, except that the literatur …
- 1,517
2
votes
Accepted
Convex optimization without Slater's condition
This is a partial answer, which addresses the practicalities of and workarounds for solving convex optimization problems not satisfying Slater's condition. It does not address the existence of a polyn …
- 1,517
1
vote
Accepted
Convexity of a positive definite objective with min(x,y)-nonlinearity
$f(x)$ is not convex. Here is a counterexample to its convexity in MATLAB notation.
C = [2 1;1 2]
x1 = [1 2]'
x2 = [2 1]'
x3 = 0.5*(x1 + x2)
Then
f(x1) = f(x2) = 8
f(x3) = 9 > 0.5*(f(x1) + f(x2))
- 1,517
2
votes
Accepted
Matrix Completion SDP Relaxation and Duality
The derivation of this dual is provided in example 8.8 "Sum of singular values revisited" of section 8.6 "Semidefinite duality and LMIs" of Mosek Modeling Cookbook 3.2.1.
- 1,517
6
votes
Accepted
If $\ell_0$ regularization can be done via the proximal operator, why are people still using...
The application of proximal gradient to this exact problem is considered in (equations (2) and (35) in) the Uncertainty in Artificial Intelligence (UAI) 2019 paper Fast Proximal Gradient Descent for A …
- 1,517
0
votes
Maximizing a pseudoconcave function in a box
You can formulate this in YALMIP (use sqrtm rather than sqrt to avoid convex modeling, which will fail, at least in this initial formulation).
If as you say, any local maximum is a global maximum, …
- 1,517
1
vote
Is this parametrized semidefinite program convex?
Yes, this is convex because the objective function and all constraints are convex.
The objective function is affine (linear), which is convex. The semidefinjite constraint on X is convex. The trace e …
- 1,517
2
votes
Quasi-concavity of $f(x)=\frac{1}{x+1} \int_0^x \log \left(1+\frac{1}{x+1+t} \right)~dt$
f(x) is increasing from 0 1o 1, where it reaches its global maximum, and decreasing from 1 to $\infty$. It is concave from 0 to 2.00841388..., and convex beyond that (note that the 2nd derivative of f …
- 1,517
1
vote
Is it possible to “solve” iterative (convex/non-convex) optimization problems via learning (...
Solve the unconstrained least squares problem in "one-shot", for example by QR or SVD (if not too big), if you consider that to be "one-shot". Then if the optimal $x$ to the unconstrained least squar …
- 1,517
1
vote
find a PSD matrix that that verify matrices sum of equality
The matrix variable $X$ only appears linearly (affinely), so this can be formulated and solved as a (convex) Linear Semidefinite optimization feasibility problem. Unless the solver runs into numerical …
- 1,517