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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5
votes
1answer
39 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
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0answers
110 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
49 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
86 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
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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
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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
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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
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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
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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 ...
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
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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 ...
2
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0answers
70 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
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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 ...
16
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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 ...
2
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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
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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
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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
184 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
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{...
3
votes
1answer
110 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 ...
1
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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
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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
160 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
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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 $p_i$ ...
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 ...
16
votes
2answers
762 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 ...
17
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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 ...
9
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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 ...
14
votes
4answers
861 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 ...
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 ...
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 ...
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
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1answer
788 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.
12
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2answers
211 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. ...