Questions tagged [quadratic-programming]
For questions on quadratic programming, methods to solve them and related solvers. Use this tag along with (optimization).
45
questions
5
votes
1answer
88 views
Solver Recommendation : Discrete Variables and Quadratic Constraints
I would like some solver recommendations to solve a problem with boolean/integer variables, mostly linear constraints but also some quadratic constraints. I also have an objective to minimize which is ...
5
votes
0answers
113 views
Water quality component optimization
I have an optimization problem that I'm attempting to tackle. As you can see in the image below, there's a graph network through which water flows. I've drawn out the problem in the image to explain ...
1
vote
1answer
50 views
Integral of PWL-Constraint in Gurobi (Java)
In my optimization model, I use piecewise-linear constraints with the output of $y[m]$. The question or problem I have now is whether there is a way in Gurobi (Java) to form the integral for this PWL ...
1
vote
1answer
87 views
Method (Algorithm) and Objective used in Gurobi Model
How can I find out which method or algorithm was used to solve my model? When I use GRB.IntParam.Method- to output the method, I get ...
5
votes
2answers
76 views
MAX-CUT: are there any algorithms or codes for classical computers, that cater to this specific case?
I missed the opportunity to ask this on OR.SE by 24 days! I asked it at CS.SE on 6 May 2019 and OR.SE entered Private Beta on 30 May 2019.
It's a problem about minimizing a sum of terms that are ...
3
votes
2answers
81 views
Implementing NLP as QP on docplex
I wanna learn how to solve non-linear programs using the docplex library. according to this link
I should be able to run the model as a QP. But when running the model I got the error: ...
3
votes
1answer
161 views
Linearizing a quadratic function with more variables or not in Gurobi?
Suppose I want to set the price $0 \le p_t \le p_{max} $ and based on the price, demand is determined $D_t(p_t)=a-bp_t$. Inventory level at each time is denoted by $I_t$ and it is determined by $I_t= ...
2
votes
0answers
38 views
Is this semidefinite constraint in fact pointless?
On Wikipedia, I encountered a statement that the semidefinite relaxation of a quadratically constrained quadratic program can be solved more efficiently (using only LP) in the case that no variable is ...
7
votes
3answers
959 views
How do Quadratic Programming solvers handle variables without bounds?
Solvers for non-convex QPs generally do the McCormick relaxation of the term $xy=z$ and then do spatial branch and bound.
This requires that $x$ and $y$ have given bounds, how do they handle the case ...
4
votes
2answers
168 views
Piecewise linear and global optimization
I am new to OR, and apologies if my mathematical notation is not clear. I have tried my best to keep it concise and given an explanation with numerical data.
I would like to understand:
Can this ...
2
votes
1answer
83 views
Minimizing a quadratic binary nonconvex function by CPLEX
I am using CPLEX 12.8 to minimize a quadratic binary nonconvex function, according to quadratic function by CPLEX.
In particular, my function is the following:
$$ \sum_{i=1}^{m-1} \sum_{f=1}^{F} \sum_{...
2
votes
1answer
51 views
Non RLT-Cutting planes for nonconvex QPs?
Consider a general nonconvex QP $x^\top Qx$. This can be linearized in an extended space by using the variable $Y=xx^\top$.
Now a valid inequality $a^\top x \le b$ can be strengthend by the RLT ...
4
votes
1answer
101 views
Linear objective function with power term in constraint
Given $n$ variables $x_{i}$ where $i\in [0,n)$, denoted as a vector $x$, given a linear objective function that we want to minimize $c^\top x$ with 2 constraints:
$\sum x_{i}^{2} < n+1$
$\sum\log(...
2
votes
0answers
71 views
How to linearize this multiplicative constraint?
I have a constraint in the form
$\sqrt{|\sum_{c\in C}{h_cw_c}|^2}\ge\sqrt{x}\zeta$
Here, $h_c$ is s row vector (know), $w_c$ is a column vector (variable).
$x$ and $\zeta$ are also optimization ...
2
votes
0answers
57 views
Determine set of “arbitrage-free” regional prices
I am seeking for a way how to determine set of "arbitrage-free" regional prices for a single commodity market.
There are $N>1$ production units with costs $C^{prod}_i, i=1,\dots,N$ and ...
5
votes
1answer
194 views
PAVA-like solution to simple QP
Let $l,u\in\mathbb{R}^n$, and consider the QP:
$$\min_{l\le x\le u} {(\Delta x)^\top (\Delta x)}$$
where $\Delta x=[x_2-x_1,\,x_3-x_2,\,\dots,\,x_n-x_{n-1}]^\top$.
I.e. we want to minimize the squared ...
2
votes
3answers
274 views
Why can quadratic functions over polyhedrons be minimized exactly in finite time?
I have heard it said that QP problems
$$\min f(x) = \frac 12 x^TAx + b^T x$$
$$x \in P$$
where $A$ is a symmetric matrix and $P$ is a polyhedron can all be solved exactly and in finite time (or it can ...
2
votes
0answers
76 views
Indicator function for integer variable with inequality constraint
I have $n$ integer variables $\vec{x}$ with the following integer programming problem.
$$
COST = \sum^{n-1}_{i = 0} a_i x_i + \sum^{n-1}_{j=0} b_j I(x_j > 0)
$$
Here, $a_i, b_j \in \mathbb{R}_+$ ...
6
votes
0answers
87 views
Cases where RLT/SDP relaxation does not work well with standard quadratic optimization
(For people who don't know what RLT is): I am maximizing an indefinite quadratic function over a standard simplex, i.e., the standard quadratic optimization problem. A well-known approach is to relax ...
10
votes
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_{...
5
votes
2answers
186 views
How to linearize a quadratic constraint to add it then via a callback function
Suppose we have a positive continuous variables $0 \le x \le UB$ where $UB$ is a known upper bound.
How can we linearize the term $x^2$?
Detailled problem:
Suppose that via a callback we compute a ...
8
votes
2answers
122 views
(Iterative?) Solutions to a certain quadratic program with non-convex constraints
Let $y\in\mathbb{R}^m$, $\tau\in\mathbb{R}$ and $X\in\mathbb{R}^{m\times n}$, with $\tau>0$
I would like to efficiently solve the following problem:
Problem 1
Choose $\alpha,z\in\mathbb{R}^m,\beta\...
3
votes
1answer
112 views
Linearizing power term in objective function
I would like to linearize $x^2$ term in my objective function. I understand this can be solved using quadratic programming solver; however, for my use case linearization is necessary to convert it to ...
3
votes
0answers
125 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{...
1
vote
0answers
80 views
Question on quadratically constrained quadratic program
If the constrained optimization problem is a quadratically constrained quadratic program of the form \begin{align}\min&\quad x^HQx-a(x+x^H)+b|z^Hx|^2\\\text{s.t.}&\quad\|x\|^2\le1\end{align} ...
3
votes
1answer
204 views
Supported pyomo free solvers for (non-convex) quadratic programming
Any one had the chance to use pyomo with free/open-source solvers that handle quadratic optimization problems, which they could be convex or not, but preferably as general as possible.
4
votes
1answer
285 views
Transforming a Quadratic constraint to SOCP
I have a problem similar to Markowitz portfolio optimization that I would like to transform into second-order cone programming. I have a linear objective function with a quadratic constraint (assuming ...
6
votes
4answers
157 views
Sequential quadratic programming source
What are the good text books to learn SQP? Are there any online courses that you can suggest?
6
votes
2answers
356 views
Assignment problem where assignments must be done sequentially
I have a weird planning problem. I think it falls under the assignment category, but I'm not sure because I'm not familiar with assignment problems, and also because there is a "temporal" angle to it, ...
7
votes
1answer
161 views
Why is there a constant in the objective function of the *best subset selection problem*?
This article presents the following formulation of the best subset selection problem
$$\min_{\|\beta\|_0\leq k}\frac{1}{2}\|y-X\beta\|^2_2$$
I wonder where the $1/2$ comes from. Help appreciated.
8
votes
0answers
91 views
For subset selection regression as a mixed integer program, how tightly should the bounding box be set?
When solving best subset regression as a mixed integer program, how do you decide how tightly to bound the range of values of the $X$ values? When the box is tight, the solver finds a solution ...
10
votes
4answers
1k views
Integer programming problem with simple quadratic objective function in Python
I have $n$ objects that need to be divided among $k$ groups. Each group must receive at least $5$ objects. In addition, the percentage of objects in group $i$ should be as close as possible to $p_i$ ...
9
votes
2answers
323 views
Convexity of a QP
In quadratic programming (QP), you encounter an objective of the following form:
$$x^TQx + c^Tx$$
and often it's desirable to know if the QP is convex. One method to check for convexity is by ...
9
votes
1answer
225 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 ...
16
votes
2answers
472 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 ...
16
votes
2answers
765 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 ...
11
votes
1answer
156 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 ...
7
votes
2answers
372 views
How can I linearize or convexify this binary quadratic optimization problem?
I have an optimization problem as below. I am having a hard time with the last constraint.
$\max \eta$
subject to
${\bf U}(:,m)^T{\bf A}{\bf U}(:,m)=0,m=1,2,\cdots,M$
here
$\bf{A}$ is a Binary ...
13
votes
1answer
790 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.
16
votes
3answers
974 views
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 ...
10
votes
2answers
144 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 ...
14
votes
4answers
871 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 ...
16
votes
2answers
228 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 ...
12
votes
2answers
214 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.
...
17
votes
3answers
346 views
How to model nonlinear regression?
As part of my research in statistics, I recently stumbled upon the paper by Wang, 2006, although its primary audience is for those who teach.
For simple linear regression, quadratic programming can ...
