# Tag Info

Accepted

### Divisibility constraints in integer programming

I going to assume that the ratio $L(x)/Q(x)$ is nonnegative. If it can be negative, I think there may be a workaround, but this will complicated enough without dealing with that. I'm also going to ...
• 39.3k

### Nonlinear integer (0/1) programming solver

Option 1: Submit as is to a solver which can globally optimize MIQPs having non-convex objective, and which might reformulate to a linearized MILP model under the hood. Such solvers include CPLEX, ...
• 13.5k

One definition of quadratization (perhaps there is more) is provided in the paper by Boros, 2018. In non-mathematical terms, quadratization is defined as a quadratic reformulation of the ...
• 5,412
Accepted

### Bin Packing with Relational Penalization

Here is a simpler symmetry-less formulation based on the one proposed by @RenaudM. For $i \le j$, let binary variable $r_{i,j}$ indicate that the bin represented by item $i$ contains item $j$. (Here,...
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### Nonlinear integer (0/1) programming solver

Maybe I am missing something but it looks like there is no need for a library: \begin{align} \sum_i \sum_j \sum_k x_{ji} y_{kj} cost(i,k)&=\sum_i \sum_j x_{ji} \sum_k y_{kj} cost(i,k) \end{align} ...
• 141

### Do convex quadratic problems always have sparse solutions?

Consider the convex QP $$\min \lbrace \sum_{i=1}^n x_i^2 : x\in \mathbb{R}^n, \sum_{i=1}^n x_i = 1\rbrace.$$ The solution is $$x = (\frac{1}{n},\dots,\frac{1}{n}),$$which is fully dense.
• 39.3k
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### CPLEX non-convex Quadratic Programming algorithms

The best publicly available CPLEX global QP algorithm description I am aware of is the tutorial presentation by Ed Klotz of IBM at the March 2018 INFORMS Optimization conference. Performance Tuning ...
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### Bin Packing with Relational Penalization

This is very related to the bin packing with conflicts problem (see eg. here), where you model the conflict as "soft" (with a binary variable to indicate violation, with a penalty in the objective ...
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### How can I model regression with an asymmetric loss function?

I suspect you want to minimize $\sum\limits_i \left[k_1\max(e_i,0)^2 + k_2\max(-e_i,0)^2\right]$. For instance, $a=0.5$ would correspond to $k_1 = 0.25$ and $k_2 = 2.25$). This can be formulated as a ...
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### Where can I find test instances for convex quadratic programming?

There are at least three more problem libraries that you can access. OR-Lib has instances of Quadratic Assignment/Knapsack/Minimum spanning tree that you can use. MINLP-Lib has several QP, BQP, IQP ...
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Rather than solving this directly as MIQCQP, you might consider linearizing the products $y_{ij} x_{ij}$, as shown here, yielding instead an MILP problem.
• 32.3k
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### How to model nonlinear regression?

NEOS has a nice web page on nonlinear least squares. It contains several classic (i.e., not so new, but still good) references for nonlinear least squares. There is a very nice introduction to the ...
• 13.5k

### Nonlinear integer (0/1) programming solver

Besides the traditional linearization suggested by @MarkL.Stone and @Richard, you might consider using the constraints to obtain a compact linearization. Explicitly, multiply both sides of your ...
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• 6,212

### How to model nonlinear regression?

From a pure optimization point of view you can say that it is possible to transform a linear fractional problem into a linear one (see for example here). In your case, this seems not feasible, as you ...
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### Where can I find test instances for convex quadratic programming?

You can reformulate Quadratic Programs as Second Order Conic Programs (SOCPs). Therefore, you can use many conic benchmark libraries and filter SOCPs. For example, Conic Benchmark Library can be a ...
• 3,980
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I would suggest reading Numerical Optimization by Nocedal and Wright. It has a pretty neat chapter devoted to SQP methods.
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### Efficient way to solve "easy" quadratic optimization problem

Fast resolution Let's start with speeding up the solution process. For the optimal solutions, all variables will be either inactive ($x_i = 0$) or, due to $\sum x_i = 1$, contribute equally to the ...
• 1,877
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### Bilinear programming vs Mixed integer linear programming performance comparison

Constraining a variable to be binary could be expressed as a quadratic constraint: $$x\in\{0,1\} \iff x(1-x)=0$$ This is often mentioned in non-convex QCQP articles to present non-convex QCQP is a ...
• 1,196

### Are there any real-world problems where quadratization helps to solve something that couldn't have been solved without quadratization?

A recent paper by Quantum Computing Inc people is showing experiments on graph partitioning where QUBO approaches lead to better results than the state of the art. Here is the paper: https://arxiv.org/...
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### Quadratic programming using CPLEX: how to check whether candidate is an extreme point?

If $\bar{y} \in \mathbb{R}^n$ is the point you are feeding, then assuming that you have verified that $\bar{y}$ is feasible to your linear constraints to begin with, then try to find a direction $d$ ...
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### Integer programming problem with simple quadratic objective function in Python

MOSEK can handle large-scale problems of a wide variety, including conic and Mixed-Integer programming problems. The Fusion API provided by MOSEK makes it straightforward to utilize the MOSEK solver ...
Accepted

### Randomly constructing a bounded ellipsoid

As noted in a comment to the original question, I have (I believe) a proof that the ellipsoid is bounded with probability 1 (assuming the pseudorandom number generator is, well, random). I've also ...
• 39.3k