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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From Quadratic to MILP?

I am playing around with some Quadratic Programs (QPs), and I want to check if my reasoning is right concerning a re-modeling from QP to MILP. So, let's consider the below QP: (QP) $\min c^T x + x^T Q ...
Matheus Diógenes Andrade's user avatar
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15 views

Problem in understanding an equation from a paper about iterative Linear-Quadratic Regulator

I'm reading a paper about iterative Linear-Quadratic Regulator (iLQR) and there are a lot of points that I don't understand. https://homes.cs.washington.edu/~todorov/papers/TassaICRA14.pdf I think ...
user900476's user avatar
1 vote
0 answers
34 views

Convex quadratic maximization over cartesian product of simplices

Suppose we are maximizing $f(x^1,\ldots,x^t)= \begin{bmatrix}{x^1}^\top & \ldots & {x^t}^\top \end{bmatrix}^\top Q \begin{bmatrix}{x^1}^\top & \ldots & {x^t}^\top \end{bmatrix}$ ...
independentvariable's user avatar
2 votes
0 answers
80 views

Does the value function of a quadratic program stay convex when adding constraints?

I am interested in the value function of a quadratic program of the form $$ v(y)=\min_x \frac{1}{2} x^\top Q(y) x, $$ subject to a linear equality constraint $$ E(y)x=d(y), $$ and a linear inequality ...
user_lambda's user avatar
2 votes
1 answer
68 views

How to show that minimizing the epsilon-insensitive loss is equivalent to a quadratic program with inequality constraints?

This question is about an optimization problem that arises in support vector regression (SVR). Suppose you have $N$ pairs $(\vec{x}_n, y_n)$ as data and would like to find a vector of weights $\vec w \...
ForceBru's user avatar
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How do I implement this convex problem in CVXPY?

I am looking to implement the following optimization problem in CVXPY. $$ \max _{x_t} x_t' \mu - \frac{\gamma}{2} x'_t \Sigma x_t - x'_t\Lambda \Delta x_t $$ where $\Delta x_t := x_t - x_{t-1}$ and $\...
Lydia's user avatar
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5 votes
3 answers
213 views

Randomly constructing a bounded ellipsoid

In a project, I am working with constraints of the following type $$ \frac{1}{2}{x}^\top Q x + q^\top x + q_0 \leq 0 $$ where I randomly generate the data by (randn...
independentvariable's user avatar
2 votes
1 answer
59 views

epigraphs for quadratic constraints

I have a constraint of the following form \begin{equation} x^{\top}x + y^{\top}y \leq t \end{equation} where x, y are vector variables and t is a scalar variable. I can augment the variables x and y, ...
Kumar's user avatar
  • 153
3 votes
2 answers
153 views

Simplest Quadratic Programming algorithm for teaching

Can anyone recommend a straightforward quadratic programming (QP) algorithm suitable for an undergraduate engineering class? I'm interested in finding an algorithm that they can easily grasp and ...
Walton P. Coutinho's user avatar
0 votes
1 answer
81 views

Simulating an integer quadratic knapsack problem

I am trying to simulate the following quadratic integer program using $\textsf{cvxpy}$: $$ \begin{array}{ll} \underset {x_1, \dots, x_K} {\text{minimize}} & \displaystyle\sum\limits_{i=1}^{K}\frac{...
UserX's user avatar
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2 votes
1 answer
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Looking for a reliable and scalable open source quadratic solver

In the convex optimization community, there has been a growing concern regarding the search for a reliable open-source solver (especially a quadratic programming solver) that can effectively deliver ...
Mohammad Namakshenas's user avatar
2 votes
2 answers
97 views

Potential methods for solving quadratic optmization problem

I am trying to solve a non-convex optimization problem with the help of sequential quadratic programming. I need to develop an algorithm inside SQP to solve this subproblem. What potential methods (...
Muhammad's user avatar
1 vote
1 answer
33 views

Non convex quadratic problem with complex variables

I am trying to solve a non convex optimization problem with the help of sequential quadric programing. My optimization variables are complex and I have expressions for gradients, hessian etc but all ...
Muhammad's user avatar
1 vote
1 answer
55 views

Quadratic optimization with non-constant coefficients

I have a series of functions (very similar to convex quadratic equations, see the first comment below) $f_1(x), f_2(x), \dots, f_n(x)$. Each of these functions touches the $x$-axis at $a_i$, which can ...
svanderk's user avatar
3 votes
1 answer
204 views

Solver for quadratically constrained mixed-integer linear programs

I have an optimization problem with vectors $x$, $y$, and $z$, where $x$ is an integer vector. My objective function is linear (i.e. $\|y\|_1$), but one of my constraints is quadratic ($x^Ty \leq z$). ...
Carol Eisen's user avatar
5 votes
2 answers
422 views

Simple OLS problem can only be solved in SCS. Is the dual infeasible?

Essentially, I am trying to solve a simple orthogonal least-squares (OLS) problem with some constraints — the coefficients must sum to $1$, no coefficient can be less than $0$, and no coefficient can ...
Pipob Puthipiroj's user avatar
3 votes
1 answer
175 views

How do I pass an objective bound to Gurobi?

I have a non-convex Quadratic Programming over unite simplex set. I have a valid lower bound on the objective function (goal is minimization problem). If I add a constraint like $$f(x)\geq lower~bound,...
Abbas Khademi's user avatar
-2 votes
1 answer
92 views

Converting a quadratic objective function in piecewise linear function

The objective function is of the form: $max$ $x^2/2+y^2/2+z^2/2$ I would like to convert it to piecewise linear function. How do I achieve that?
scouse_s's user avatar
1 vote
1 answer
106 views

How to model and solve such a 0-1 programming problem

My problem is described in this picture(It's like a Pyramid structure): The objective function is below: $$\min\sum_{k=1}^\ell\sum_{i=0}^{2^k-1}\sum_{j=0}^{2^k-1}\left(A_{i,j}^k-A_{i,j+1}^k\cdot\frac{...
happy's user avatar
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3 votes
2 answers
930 views

Is solving a quadratic programming optimization problem using python slower than C++?

I am using the cvxpy library in python to solve a quadratic programming problem and the solver used is scip. I found that when the amount of data becomes large, the solution process will be ...
happy's user avatar
  • 63
2 votes
1 answer
357 views

MIQP — CVXPY unable to treat summation of variables as a variable

I have a quadratic integer programming assignment problem. The goal is to assign riders seats on a bus such that distance between any two riders is maximized; however, the importance of each objective ...
jbuddy_13's user avatar
  • 501
3 votes
0 answers
117 views

Linearize objective function with non-linear terms

I have a problem with linear constraints but in the objective function I want to have some linear terms along with a $x^2$ term. So it is like the following: $$\min \sum \limits _i \sum \limits _j (a[...
christouandr7's user avatar
1 vote
2 answers
256 views

How to solve this mixed integer quadratic program using cvxpy or other method?

My problem is described in this picture: $$ \begin{array}{l} \left\{\begin{array}{l} \text { objective function: } \\ f = \min \sum_\limits{l=1}^2 \sum_\limits{i=0}^{2^l-1} \sum_\limits{j=0}^{2^l-2}\...
happy's user avatar
  • 63
2 votes
1 answer
258 views

Poorly conditioned quadratic programming with "simple" linear constraints

I have many quadratic programming problems of the following form: $$\min_{x\in\mathbb{R}^n} { \tfrac{1}{2} {\lVert Cx-d \rVert}^2} $$ $$\textrm{s.t.}\ x_1\le 0,\ x_n\le 0,\ x_n\le a_1^\top x_{1:n-1},\ ...
cfp's user avatar
  • 235
3 votes
2 answers
368 views

How to view pause and view current solution in CPLEX Optimization Studio?

I am solving my first model in CPLEX 22.1. I have setup a quadratic MIP with 100 variables and the model has been running for a day already with the best integer and best bound solutions barely ...
twaits791's user avatar
4 votes
0 answers
140 views

Analytical solution of constrained quadratic program

I'm trying to solve a "simple" (= small) optimization problem often, with only minor changes to the objective function. Therefore it's important to keep the "time per solve" as low ...
kchnkrml's user avatar
4 votes
1 answer
470 views

Is this a non-linear integer model?

Let's say if I have two decision variables, $f$ and $g$ respectively, where $f$ is continuous, and $g$ is binary. If I have a constraint like this, $$ f\cdot g \le C$$ Does this make my model ...
overboxed's user avatar
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1 vote
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Does Gurobi solve QCQMIPs with Quadratic terms faster with then Bi-Linear terms in general?

Based on the color distance function defined here i try to find $n$ RGB colors with large inter set color distances and good color distance to white. ...
worldsmithhelper's user avatar
6 votes
2 answers
200 views

When should we avoid linearizing a quadratic term?

Some solvers like Gurobi can handle mixed-integer quadratically-constrained quadratic models regardless of their nonconvexity. I have some experience that Gurobi can handle instances of the max $k$-...
Ramin Fakhimi's user avatar
10 votes
1 answer
372 views

Sensitivity analysis of QP

Given a quadratic program $$ f^* \equiv x^\top Q x + b^\top x \\ x \geq 0 \\ A^\top x = d \\ x \in \mathbb{R}^n $$ I would like to analyze the sensitivity of the solution $x^*$ to perturbations in $Q$ ...
ntrstd11's user avatar
  • 235
2 votes
2 answers
224 views

Difference between constraint formulation and performance

I am wondering about the characteristics and performance of some constraints with only binary variables. I assume that solving (integer) linear programs is faster than quadratic ones. At first: $$ a,b,...
Mike's user avatar
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1 vote
0 answers
227 views

Converting quadratic constrains to linear constraint [closed]

I try to convert a quadratic constraint to a linear one: $$ w_j = \sum w_\text{j,i} \\ w_\text{j,i} = \frac{w_j}{D} \times u \\ w_j,D,u \in \mathbb{N} \\ $$ The values for $w_j$ and $D$ are constant ...
Mike's user avatar
  • 147
1 vote
0 answers
65 views

Dual of a quadratic constraint

This is my model. \begin{align} \min_x&\quad\sum_{e\in E} X_e p_e \\ \text{s.t.}&\quad\sum_{e \in E: T(e)=i} X_e - \sum_{e \in E: H(e)=i} X_e = \begin{cases}1, \;\text{if}\;i=s\\-1,\;\text{if}...
orpanter's user avatar
  • 507
5 votes
3 answers
152 views

What is the go-to practical method for optimizing separable quadratic programs?

I have a quadratic program that looks like this: For fixed vector $b$, and matrices $A_1, ..., A_n$, Find column vectors $x_1, ..., x_n$ that minimize $\sum_{i=1}^n x_i ^T 1 1^T x_i$ subject to $\sum_{...
AspiringMat's user avatar
2 votes
1 answer
492 views

Can docplex solve a mixed integer quadratic programming (MIQP) problem?

I am trying to solve a mixed integer quadratic programming (MIQP) problem. The objective function contains the product of two continuous decision variables, some of constraints are non-linear too. I ...
GTek's user avatar
  • 307
5 votes
1 answer
184 views

How to handle a non-separable bilinear objective function in the special case of decoupled constraints?

I have a large number of (10000+) non-negative, real decision variables $x_i$ and $y_j$. Let $I$ and $J$ be the index sets associated with $x$ and $y$, respectively. Let $\bar{I}$ and $\bar{J}$ be non-...
Displayed_Name's user avatar
5 votes
1 answer
371 views

How to handle a bilinear objective function in the special case of decoupled constraints?

I have decision variables $x_i$ and $y_j$, real positive variables. I would like to minimize objective function \begin{aligned} \min \quad & \sum_{ij} x_iy_j \\ \end{aligned} All constraints are ...
Displayed_Name's user avatar
3 votes
1 answer
175 views

Translate standard weighted least square regression to quadratic programming

Sorry if this is really easy for you gurus. I'm trying to derive the reformulation of a weighted least square regression to a quadratic programming form. I understand there is a closed form solution ...
inf's user avatar
  • 129
11 votes
5 answers
3k views

A Stack Overflow user's curious problem of maximising unsortedness

User ddofborg posted on Stack Overflow a programming question which hides a combinatorial optimisation problem. The idea is the following: given a list of URLs with their respective domain names, he ...
Alberto Santini's user avatar
3 votes
1 answer
67 views

Maximizing the number of nonnegative coordinates of $Wx$

I want to find good incumbent solutions to the following problem: $\newcommand{\RR}{\mathbb{R}}$ $\newcommand{\norm}[1]{\left\Vert#1\right\Vert}$ Given a matrix $W \in \RR^{m \times n}$, find the ...
Daniel Paleka's user avatar
4 votes
1 answer
292 views

Can you calculate the mean of some MIP variables using linear constraints?

got a lingering question from a graduate course in integer programming that's been bugging me ever since. Is it possible to find the mean of some variables in a MIP without resorting to quadratic ...
gjgutier545's user avatar
2 votes
1 answer
357 views

CPLEX solver for quadratic problems with barrier algorithm

I'm writing a convex minimization model with a quadratic objective function and linear constraints in C#. I set CPLEX to solve the problem with the barrier algorithm. The interesting thing is when I ...
Saba Kiani's user avatar
2 votes
1 answer
118 views

Linearize product of $x\cdot y \text{ with } x,y \in \{-1,0,1\}$ for MILP

I have a problem where I have many products between variables drawn out of $\{-1,0,1\}$. Could you suggest a linearization in terms of variables in $\{-1,0,1\}$ or $B_1 - B_2$ where $B_i \in \{0,1\}$ ...
worldsmithhelper's user avatar
3 votes
0 answers
95 views

Automatic quadratization of constraints in pyomo for gurobi

Gurobi 9 can solve QCQPs, and QCQPs capture all of polynomial optimization by the obvious trick that e.g. a cubic term $x_1 x_2 x_3$ can be turned into a quadratic term $y x_3$ and a constraint $y = ...
Martin Koutecký's user avatar
7 votes
1 answer
1k views

Is there any open source quadratic programming solver with C# API

I have a quadratic programming model (i.e., quadratic objective function and linear constraint) and, I want to solve it on an open-source solver. Since our project developed on C#, we also would like ...
cyy's user avatar
  • 73
3 votes
1 answer
98 views

optimizing quadratic form over bounded polytope

As a followup to this question, I am looking for references for the problem of maximizing $x^TQx$, where $Q$ is positive definite, subject to linear equality and inequality constraints bounding all ...
sayda's user avatar
  • 131
2 votes
2 answers
137 views

How to use `cplexAPI` to solve quadratic programs/quadratically constrained linear programs?

I am trying to use cplexAPI to solve quadratic programs or quadratically constrained linear program. While it seems to be pretty straightforward to use ...
wayne's user avatar
  • 21
2 votes
1 answer
79 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 ...
sayda's user avatar
  • 23
4 votes
1 answer
166 views

Continuous minimax with linear objective and constraints

How to solve the following minimax problem quickly? The variables are all continuous. $$\max_{x_{1}, x_{4}, x_{5}} \min_{x_2,x_3} \vec{c}^{\intercal} \vec{x}$$ subject to the following constraints: $$...
Qurious Cube's user avatar
7 votes
1 answer
430 views

Bilinear programming vs Mixed integer linear programming performance comparison

I know that both bilinear programming and mixed integer linear programming are NP-hard. But is there a preference to have when choosing an approach to solve a problem that can be represented in both, ...
TUI lover's user avatar
  • 173