# Questions tagged [quadratic-programming]

For questions on quadratic programming, methods to solve them and related solvers. Use this tag along with (optimization).

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### Bin Packing with Relational Penalization

There are $N$ bins with equal capacity $C$. In addition, there are $N$ objects $x_1, x_2, \dots, x_N$ that need to be accommodated using the least amount of bins. Each object $x_i$ has a volume ...
3answers
432 views

### How to model nonlinear regression?

As part of my research in statistics, I recently stumbled upon this paper1 which provides an operational perspective into linear models. In simple linear regression, quadratic programming can be used ...
2answers
946 views

### Divisibility constraints in integer programming

In the study of a certain pure mathematical problem (related to infinite-dimensional Lie algebras) I found myself in a situation where it would be very desirable to be able to solve an integer ...
2answers
527 views

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

The closest thing I know is the computer vision problem, in which an image is de-blurred and/or de-noised by quadratizing a quartic problem into a quadratic optimization problem (QUBO) and then the ...
2answers
270 views

### Can we replace a binary variable with a continuous variable using a quadratic equality constraint?

Is it possible to replace a binary variable $x$ with a continuous variable that satisfies the quadratic equality constraint $x^2 - x=0$? The function $f(x) = x^2 -x$ is not a convex function. Can ...
4answers
1k views

### CPLEX non-convex Quadratic Programming algorithms

CPLEX solves non-convex quadratic problems to global optimality with a global optimality option (in version 12). The relevant pages are this and this. I benchmarked many solvers, and see that CPLEX ...
1answer
853 views

### What is quadratization?

In the context of discrete optimization, what exactly does it mean to "quadratize" a function? The term seems to be used mainly by operations researchers, in my experience.
2answers
256 views

### Where can I find test instances for convex quadratic programming?

I am looking for (sources of) convex quadratic programming instances with linear constraints. I am open for both continuous and mixed integer problems, but do not want randomly generated instances. ...
1answer
441 views

### Efficient way to solve "easy" quadratic optimization problem

The linear program \begin{align} \min &\sum_{i=1}^nc_{i}x_{i}\\\ \mbox{s.t.:}&\sum_{i=1}^nx_{i}=1,\\\ &x_{i}\geq 0,&&\forall i=1,\dots,n \end{align} has a trivial optimal solution ...
1answer
159 views

### Quadratic programming using CPLEX: how to check whether candidate is an extreme point?

I am currently solving an indefinite quadratic program with linear constraints using CPLEX. Moreover, I am trying to determine whether the candidate point CPLEX is feeding my callback function is an ...
6answers
2k views

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

I have the following optimisation problem.\begin{align}\max&\quad\sum_i\sum_j\sum_k x_{ji}y_{kj} \operatorname{cost}(i,k)\\\text{s.t.}&\quad\sum_j x_{ji}=1\quad\forall i\\&\quad\sum_k y_{...
4answers
1k views

### Integer programming problem with simple quadratic objective function in Python

I have $n$ objects that need to be divided among $k$ groups. Each group must receive at least $5$ objects. In addition, the percentage of objects in group $i$ should be as close as possible to $p_i$ ...
2answers
150 views

### Global optimality condition of non-convex quadratic programs

We know that a convex quadratic maximization (not minimization!) on a polyhedron has its global optimal value on a vertex. Also, I have read in some papers that checking whether a vertex is globally ...
1answer
254 views

### How can I model regression with an asymmetric loss function?

Mosek provided a concrete example of using the Huber loss function, Huber loss, which is great! One problem I am trying to tackle is to use asymmetric loss, as described in the answer of asymmetric ...
2answers
335 views

### Convexity of a QP

In quadratic programming (QP), you encounter an objective of the following form: $$x^TQx + c^Tx$$ and often it's desirable to know if the QP is convex. One method to check for convexity is by ...
2answers
186 views

### Inverse Ising problem

Inverse Ising Problem The inverse ising problem means fitting the coupling $J_{ij}$ and field $h_{i}$ parameters given a sample of configurations of spins. Each spin $s_{i}$ is either +1 or -1. The ...
2answers
124 views

0answers
129 views

### SDP relaxation with greater-than and less-than inequalities at the same time

I am dealing with the following nonconvex fractional quadratic optimization problem \begin{align} & \min_{\boldsymbol{x}} && \max_{t \in \mathcal{T}} \frac{\boldsymbol{a}_t^T \boldsymbol{...
3answers
316 views

### Why can quadratic functions over polyhedrons be minimized exactly in finite time?

I have heard it said that QP problems $$\min f(x) = \frac 12 x^TAx + b^T x$$ $$x \in P$$ where $A$ is a symmetric matrix and $P$ is a polyhedron can all be solved exactly and in finite time (or it can ...
1answer
70 views

### Convex function subject to $0\le x_1\le \ldots \le x_n\le 1$ and linear constraint

I am maximizing a convex function (a positive definite quadratic form, if it makes a difference) subject to $0\le x_1\le \ldots \le x_n\le 1$ and a linear constraint $a^\top x+b=0$. Can I conclude ...