9
votes
Is there a high quality nonlinear programming solver for Python?
We recently released (2018) the GEKKO Python package for nonlinear programming with solvers such as IPOPT, APOPT, BPOPT, MINOS, and SNOPT with active set and interior point methods. One of the issues ...
- 506
5
votes
Accepted
Gradient of function after renormalization of variables
Based on your current optimization strategy, it is likely you cannot get away with just the gradient of $f(\cdot)$ evaluated at $\frac{\boldsymbol{x}}{\left| \boldsymbol{x}\right|}$ since ...
- 4,308
5
votes
MINLP with GEKKO - Modeling discrete variables
If they don't have a variable that is constrained to a discrete set you can formulate it like so:
...
- 4,021
4
votes
Accepted
Optimisation of purely integer quantity with bound-constraints for a 1D expensive function whose analytical form is not available
@Stellos already gave the correct answer, but let me try to back it up with a bit of intuition:
Think of your function $f(x)$ as a function that would actually make sense for any real-valued argument ...
- 56.2k
4
votes
Accepted
Example Problem to Demonstrate BiCGStab
Check out Matrix Market. It has a nice collection of matrices from different areas (fluid dynamics, elasticity, acoustics etc.)
So you do not need to assemble systems yourself in order to provide ...
- 901
4
votes
Accepted
Largest hypercuboid inside a polyhedron
It turns out that the problem has quite an elegant solution.
Let the hypercuboid be defined by $\mathbf{l} \leq \mathbf{x} \leq \mathbf{u}$ instead of using the more cumbersome $\mathbf{x_{0}}$ and $\...
- 572
4
votes
Accepted
Constraints 'exactly/at most one non-zero element' without binary variables
No, this is not possible. There is a standard way of showing this:
The feasible region of your constraints is not convex. For example, $x_{1,1}=1$, $x_{1,2}=0$ is feasible, $x_{1,1}=0$, $x_{1,2}=1$ ...
- 18.9k
4
votes
Accepted
MINLP with GEKKO - Modeling discrete variables
Below is an example of using Gekko (v0.2.4+) to define a SOS1 variable with an objective function to find the minimum in that sequence of values.
...
- 506
3
votes
Accepted
Piecewise-Linear Quadratic Optimization for an "Almost Convex" Problem
You could implement the piecewise linear functions using SOS2 variables (SOS2=Special Ordered Sets of Type 2). This approach does not care about convexity (that is, convexity related to the piecewise ...
- 271
3
votes
Accepted
Why can't we discretize continuous domains in distributed non-convex constraint optimization problems?
You are missing that, in general, integer optimization problems are much more expensive to solve than real-valued optimization problems. That's because for real-values optimization problems, you have ...
- 56.2k
3
votes
Accepted
SOCP: Recovering primal from dual
The standard technique is to solve the optimality conditions as a
linear system of equations. Let us rewrite your problem into standard
form
$$
\begin{align*}
\text{maximize } & -c^{T}x\text{ s.t. ...
- 2,495
3
votes
Accepted
Minimizing linear objective on intersection of convex sets
See this recent paper on an extension of stochastic gradient descent that could be used on your problem:
https://arxiv.org/abs/1511.03760
You could also apply Dykstra's algorithm (or any other ...
- 18.9k
3
votes
Accepted
Symmetric nonnegative matrix factorization
I'm going through some of my old StackExchange posts and came across this one. As it turns out, the answer led to a section in a published paper!
As detailed in that paper (and in its notation), if ...
- 1,423
3
votes
Geometric Programming - symbolic version
You may want to give SCAT Maple package a try. It is certainly not tailored to geometric programming but is worth trying.
C. Hamilton "Symbolic Convex Analysis" thesis from 2005 describes the ...
- 8,712
3
votes
Accepted
Setting up optimization problem in GEKKO
A good first step with any parameter estimation problem is to solve it in simulation to verify that you can get a good solution and that the parameters have an effect on the objective. You can first ...
- 506
3
votes
What's the right choice of variable settings for setting up my optimal control problem?
The m.MV() type has additional tuning parameters such as move suppression that is likely contributing to the difference in solution. Also, the ...
- 506
3
votes
A question related with $p$-Laplacian and conjugate gradient method
You seem to be misunderstanding the difference between the derivative of $E$ and the gradient of $E$. I am going to assume that $u$ lies in an appropriate Sobolev space $V$, which embeds continuously ...
- 2,936
3
votes
optimizing piecewise linear objective functions (perhaps non convex) with equality constraints
For global optimization of black box functions, I have successfully used scipy's differential evolution (https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.differential_evolution.html)...
- 41
2
votes
Is there a high quality nonlinear programming solver for Python?
KNITRO has Python and MATLAB interfaces, among others. Think of it as an FMINCON replacement, but much better performing, and more expensive. https://www.artelys.com/en/optimization-tools/knitro#doc-...
- 2,242
2
votes
Accepted
Backtracking-Armijo Line Search Algorithm
[The following references a paper by Ascher et al. ...]
You want to solve
$$
\partial_t f(x,y,t)= (\partial^2_x +\partial^2_y) f(x,y,t),
$$
subject to some initial and boundary conditions. However, ...
- 2,226
2
votes
Optimisation of purely integer quantity with bound-constraints for a 1D expensive function whose analytical form is not available
Without any information/insights about $f(\cdot)$, your only option to find the maximum is by brute force, i.e., evaluate $f$ for all integers. Now, if this is is practically impossible to do, your ...
- 741
2
votes
Accepted
Obtaining a feasible solution for underdetermined system of linear equations satisfying inequality constraints
The problem was already solved in the comments section using linprog. Also CVX...
I wanted just to point out that fmincon can also be used: just set the linear ...
- 1,077
2
votes
Perturbation in bounds given the perturbation to constraints
I assume that you're interested in
$b-\epsilon \leq a^{T}x \leq b +\epsilon$
rather than
$b+\epsilon \leq a^{T}x \leq b +\epsilon$
right?
If you're interested in more complicated ...
- 18.9k
2
votes
Accepted
Slightly change two vectors to satisfy a constraint
There are of course infinitely many vectors $\vec \alpha,\vec \beta$ that satisfy $\vec \alpha\cdot\vec \beta=c$. So if you want to have a particular pair of vectors, you will have to be precise when ...
- 56.2k
2
votes
Why does Newton's method with Linear Equality Constraints use KKT condition?
You know that you have to use the Lagrange multiplier,
$$
L(x,\mu)=f(x)+\mu^T(Ax-b).
$$
The derivatives of that function that have to be zero at a possible minimum are
$$
0=\frac{\partial L}{\partial ...
- 6,149
2
votes
Solving a parameter estimation problem using trajectory optimization
I am a bit confused as to your characterization of constraints. Equation $(1)$ is not a constraint. It is the model that generated the time series data you are trying to fit. You then try to find the ...
- 2,936
2
votes
Solving a parameter estimation problem using trajectory optimization
Your cost function can also be written as
$$
K = \int_0^{t_f} \left(\phi(t) - e^{-M^\top \tilde{D}\,M\,t} \hat{\phi}(0)\right)^\top \left(\phi(t) - e^{-M^\top \tilde{D}\,M\,t} \hat{\phi}(0)\right) dt....
- 470
2
votes
Applying displacement control loading using lagrange multipliers in the material non-linear finite element method
The addition of the Lagrange multiplier term is an unnecessary complication in this case, so let's just consider the simpler root finding problem
$$
F(u) = \frac12 u^TK(u)u - fu = 0.
$$
Your ...
- 56.2k
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