14 votes
Accepted

Is there a constrained nonlinear optimization library like IPOPT that runs on GPUs?

tl;dr: My general impression from the literature is that speedups are modest (if they exist). The main kernel you'll see in these methods is a sparse-direct method (e.g., sparse LU, sparse LDLT), and ...
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8 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 ...
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7 votes
Accepted

Piecewise linear optimization with resource allocation constraints

If $P_i(w)$ is a piecewise linear, convex function, then it can be written as the maximum of a number of linear functions, $P_i(w)=\max \{L_{i1}(w),\ldots,L_{iJ_i}(w)\}$. Then, the optimization ...
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6 votes
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How to solve this optimization problem with abs object function?

It looks like you are optimizing over the surface of a ball. That constraint is non-convex, but I think with your objective the solution will always be on the surface, so you can relax it to be $ \leq ...
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5 votes
Accepted

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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  • 505
5 votes
Accepted

0,1 binary polynomial programming

If the $s_{i}$ are integers, there are reformulations of integer polynomial terms that result in mixed-integer (linear) programs, at the cost of introducing additional variables and constraints. Fred ...
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5 votes

minimization of normalized constrained quadratic function

A good starting point for this is Boyd and Vandenberghe's textbook, Convex Optimization. I'd strongly encourage you to get a copy (the authors have posted a free .pdf online, so it won't cost you ...
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5 votes
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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 ...
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  • 3,673
4 votes
Accepted

Does the amount of correlation of model parameters matter for nonlinear optimizers?

If possible, you want to choose a well-scaled objective for your problem, which would mean de-correlating your parameters as much as possible. For a quadratic approximation to an objective function, ...
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4 votes

Optimization with matrix determinant as constraint

Replace $A$ by a factored form from which the determinant can be computed stably and cheaply. E.g., add new triangular variables $L$ and $R$ and the constraint $A=PLR$ for some fixed permutation ...
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4 votes

How can I solve a nonlinear optimization problem where constraint contains exponential term?

If the $R_k$ are your only optimization variables, then the constraint $$ \exp \left[ - (2^{{R}_k } -1) \left(\frac{\tilde{Z} g_{k} p_{\max} + \sigma^2}{g_{pu} p_{pu}} \right) \right] \leq q $$ is ...
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4 votes

How to solve a constrained optimization problem using minFunc or minConf

Not a direct answer to your title question, but I think you are better off attacking this problem from the semidefinite domain instead. Trivial approach is to linearize the objective at some initial ...
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4 votes

How to solve a constrained optimization problem using minFunc or minConf

Adding another answer, as I just realized that the problem is easily solved as a linear SDP. Let $Q=S^TS$ and you have the objective $\mathrm{trace}~Q + \mathrm{trace}~Q^{-2}$. Introduce an upper ...
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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 ...
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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 $\...
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  • 562
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 ...
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  • 861
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, $x_{1,1}=0$, $x_{1,2}=1$ ...
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4 votes
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How to solve the following Frobenius norm-minimization problem?

$$\big\| \mathrm A \begin{bmatrix} \mathrm X & \mathrm X^2\end{bmatrix} - \begin{bmatrix} \mathrm B_1 & \mathrm B_2\end{bmatrix} \big\|_{\text{F}}^2 = \underbrace{\| \mathrm A \mathrm X - \...
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4 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: ...
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  • 3,111
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. ...
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3 votes
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GAMS solvers: which one to use

Somewhere in the GAMS file, after you've declared almost all of your model, you have to write a solve statement of the form ...
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3 votes
Accepted

Find $\min \sum_{1\le i\le n} x_i\mathbf{z}^T\mathbf{A}\mathbf{y}_i +\mathbf{b}^T\mathbf{x} +\cdots$

I'm looking forward to your comments to the following solution. We will solve the problem for the constraints $\mathbf{1}^T\mathbf{x}=\mathbf{1}^T\mathbf{y}_i=\mathbf{1}^T\mathbf{x}=1$ and for a ...
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  • 505
3 votes

Is there a high quality nonlinear programming solver for Python?

How about calling Matlab from Python, using python-matlab-bridge or the like ? That looks much easier than porting yards of code, not to mention test cases and doc. And general: any Python $\to$ any ...
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  • 892
3 votes
Accepted

Large-scale nonlinear optimization problem

At 1 million variables, you're sort of on the cusp of what a code like (the publicly released version of) IPOPT can do. IPOPT will solve the KKT system derived from an interior point method using ...
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3 votes

A separable nonnegative quadratic program

With $d=16$, the $Q$ matrix is just 256 by 256. Thus the individual subproblems are quite small. Your $Q$ matrices are singular, so the optimal solution to each of the subproblems is likely to be ...
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3 votes

Optimization with matrix determinant as constraint

The constraint $\det(A) > 0$ is very unstable and absolutely horrible. Would you believe that on practical, non-contrived 6 by 6 symmetric matrices, changing one element of the matrix by 1e-15 can ...
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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 ...
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3 votes

Is there a constrained nonlinear optimization library like IPOPT that runs on GPUs?

I'm a little late to the party, but the short answer is that yes it's possible to parallelize an interior point method for GPUs, but whether or not that is successful depends on the structure of the ...
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  • 677
3 votes
Accepted

Sum of Inverse of Variables in an Optimization Problem

For the discrete version, it can be cast as a mixed-integer linear program. You just have to note that every element $x_i$ can be written as $x_i = \sum_{j=1}^k \frac{\delta_{ij}}{j}$ where $\sum_{j=1}...
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