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Questions tagged [quadratic-programming]

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0
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1answer
16 views

Minimize squared error of linear function

Let $M$ be a $m \times n$ matrix, $x$ a $n$-vector, $y$ a $m$-vector, and $\|\cdot\|_2$ represent the $L_2$ norm (i.e., Euclidean norm). Given $M,y$, the goal is to find $x$ that minimizes the ...
2
votes
1answer
80 views

Checking positive definiteness on a hyperplane

Is there a faster way to check whether $A\in\mathbb{R}^{n\times n}$ is positive definite on $b^{\bot}:=\{x\in \mathbb{R}^{n}: x\cdot b=0\}$ than ...
3
votes
1answer
39 views

Improve optimization speed for a set of similar problems: Quadratic programming with a warm start

I am repeatedly solving quadratic program, $x^T Q x$ with time dependent linear constraints $Ax=b_t$. Dimension of $x$ is around 10000 and there are around 50 constraints. I want to solve the ...
2
votes
1answer
95 views

What is the fastest way to solve Ax=b (subject to constraints and an absolute term)

I am trying to solve/optimize $Ax=b$ in the least squares sense subject to box constraints; a few (less than 5) equality/inequality constraints; and an absolute function penalty (or some other ...
0
votes
1answer
436 views

How to define the derivative for Scipy.Optimize.Minimize

I am trying to use scipy.optimize.minimize to minimise a quadratic objective function: $f(x) =x^\top Q x$. As a start, I have successfully implemented this using the built-in Nelder-Mead Simplex ...
1
vote
1answer
159 views

Reformulate a strictly convex QP problem containing absolute value term

Can the following strictly convex optimization problem be reformulated into a standard form that is also a strictly convex problem? $$\begin{align} &\text{Minimize }\frac{1}{2} x^T Q x + a^T x + ...
6
votes
0answers
148 views

Multi-matrix orthogonal basis problem

Suppose we are given a set of symmetric, positive definite matrices $A_1,A_2,\ldots,A_k\in\mathbb{R}^{n\times n}$. Is there any numerical method or reduction to a known problem (e.g. eigenvalue ...
3
votes
2answers
96 views

Approximating solutions to quadratic recurrence boundary value problem

Cross-posted from Math Stackexchange: https://math.stackexchange.com/questions/2421964/approximating-solutions-to-quadratic-recurrence I have a branching process problem that has been reduced to ...
0
votes
2answers
117 views

Factoring a quadratic function

I have a quadratic binary optimization problem of the form \begin{align} &\max x^TQx \cr &\text{subject to }x\in\mathcal{X}\subseteq\{0,1\}^n, \end{align} where $\mathcal{X}$ is the feasible ...
1
vote
1answer
150 views

How can I solve on a computer a large projection problem with redundant constraints?

This question is the essence of this one. After we remove all the cruft, we can recast it as follows: Problem: Given $b \in \mathbb{R}^n$, $C\in \mathbb{R}^{n\times m}$, and $g\in \mathrm{Range}(C^...
3
votes
1answer
71 views

Solving multiple least-square problems with the same constraints

The following least-square problem can be solved efficiently (e.g. using matlab's lsqlin): $$\vec{x}^*=\arg\min_\vec{x} ||C\vec{x}-\vec{t}||^2\,\ s.t.\ Ax \le \vec{b}$$ where the parameters of the ...
4
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2answers
400 views

Quadratic programs with rank deficient positive semidefinite matrices

Let $A$ be a $n\times n$ square symmetric matrix. In addition, $A\succeq0$ and $\mathrm{rank}(A)<n$. This means that all eigenvalues are non-negative, but also that there are some zero eigenvalues. ...
1
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0answers
53 views

energy computation for BVP with Dirichlet boundary conditions

I am solving quadratic minimization problem \begin{align} \min_{x}\ \frac{1}{2} x^T A x -b^T x, \end{align} where matrix A results from discretization of Laplacian by FEM method, subjected to ...
1
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0answers
44 views

minimization a quadratic form with linear constraints (Prospective method SMIC 74 method)

my problem is a ecuation i don't understand, I have no idea how to solve that, is a classic minimisation programme of quadratic form with linear constraints, here is the ecuation: $$\sum_{i,j}^n[P(i/...
2
votes
2answers
174 views

Approach to handle a quadratic constraint xy <= z

I have non-linear constraints like $ x_1x_2\leq x_3 $ where $ x_1,x_2,x_3\geq 0 $. The objective is linear, and all other constraints are linear, too. I know that I can transform the product as $ ...
0
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1answer
128 views

Optimization with Yalmip [closed]

I would like to solve in Matlab the following optimization problem $$\begin{array}{ll} \text{maximize} & \bigg\| \displaystyle\sum_{l=1}^{2}\alpha_l \int_{\tau_{m+l-1}}^{\alpha_1\tau_m+\alpha_2\...
3
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0answers
64 views

“Solution path” for quadratic program as regularizer changes

I am solving a quadratic program with regularization parameter $\alpha\geq0$ to get the solution to a problem of the form $$ p(\alpha):= \arg\min_{p\in\mathbb{R}^n}\ [\alpha(v^\top p)+f(p)], $$ where $...
1
vote
1answer
139 views

What is the fastest method for solving a quadratic programm repeatedly,( warmstarted)?

I would like to solve the following optimization problem \begin{align} \min_{x\in [0,1]^n} x^T p+ \frac{1}{2\lambda} x^T Q x \end{align} $Q$ is a positive semidefinite matrix. $\lambda>0$ is a ...
1
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0answers
44 views

Admissible box constraint for a quadratically constrained linear program

I am looking at a real-world resource allocation problem that is cast as a quadratically-constrained linear program of the form $$ \max\langle f,x\rangle $$ subject to $$ \begin{aligned} m \leq\,\, &...
1
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1answer
131 views

How to efficently solve: min $\sum_{ij}(a_{ij}x_{ij}^2 + b_{ij}x_{ij})$ s.t

I am trying to solve the following problem, where $a_{ij} \ge 0 \ \forall i,j$: \begin{align} \mbox{minimize}\quad & \sum_{i=1}^m\sum_{j=1}^n (a_{ij}x_{ij}^2 + b_{ij}x_{ij})\\ \mbox{subject to}\...
1
vote
1answer
187 views

Reformulation of optimization problem

I am looking for some helps concerning an optimization problem. I have an optimization problem defined on two sets $\mathcal{X}=\{x_i\}_{i=1}^n $ and $\mathcal{Y}=\{y_j\}_{j=1}^m $ and described as ...
6
votes
1answer
707 views

Algorithm for solving system of quadratic equations and linear equations

Let $x \in R^N$. From a Spectral Chebyshev collocation method, I have a system of quadratic and linear equations. Denote them, $$ x^T Q_i x + L_i^T x = 0 $$ and $$ A x = 0 $$ Furthermore, I know ...
3
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0answers
142 views

Eigenvalue-style optimization with quadratic constraints

Suppose $A\in\mathbb{R}^{n\times n}$ is symmetric and positive definite and that we have several symmetric matrices $B_i\in\mathbb{R}^{n\times n}$ that are low-rank and indefinite. I need an ...
5
votes
2answers
1k views

How to determine whether two cylinders intersect or not?

Considering any two cylinders, defined as: the center of their bottoms $A_i$, the radius of their bottom $R_i$, the unit vector $W_i$ of their axis direction, and the length $L_i$ of the cylinders, ...
1
vote
1answer
230 views

How to formulate variance minimization as a mixed integer quadratic program

I have a mixed integer quadratic problem and my objective function is as follows $$\arg \min \operatorname{Var}(f(x),g(x)) + \operatorname{Var}(c(x),d(x)) + \cdots$$ where $f$, $g$, $c$ $d$ are ...
0
votes
1answer
76 views

Can Variance be replaced by absolute value in this optimization problem

Initially I modeled my objective function as $$\arg \min \operatorname{Var}(f(x),g(x)) + \operatorname{Var}(c(x),d(x)) + \cdots$$ where $f$, $g$, $c$, $x$ are linear functions. To be able to solve ...
3
votes
1answer
2k views

Convex Optimization problem with sum of absolute value constraints

How to solve the optimization problem written below? $$\begin{align} &\operatorname{argmax}\limits_{a}\; a^T b - \frac{1}{2} a^T X a\\ &\text{subject to } \sum_i |a_i|=4,\; \sum_i a_i = 0 \...
2
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0answers
83 views

Disjunctive programming software

Can you advise me any kind of existing software that can help to solve the disjunctive programming problem? The problem is the following. We have unit 3D planes $\Pi_{1}, \ldots, \Pi_{N}$ (they are ...
2
votes
1answer
142 views

Solve $AX = B$ where $X^T X = C$

Is there a natural way to find the solution to $$AX = B, X^TX = C \enspace \text{?}$$ $X$ is a matrix and has a small number of rows, and $A$ is sparse. An approximate solution would be fine.
1
vote
2answers
737 views

Solving absolute value quadratic optimization problem

would you please help me to solve following problem $$x^*= \text{argmin}\ xLx^T+ |P^Tx|$$ $x$ is binary $P$ is a known vector (with positive and negative values) $L$ is Laplacian matrix I have ...
0
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2answers
186 views

Convex quadratic problem solver gives different answers?

I am pretty sure that the following variance objective function should be a convex quadratic problem. My objective function is as follows: $$ \text{argmin } \text{var }(X*L) \xi \geq 1, \text{ where }...
4
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1answer
1k views

Converting convex quadratic constraint to linear matrix inequality (LMI)

I have the quadratic programming problem in $x$ $$\text{Minimize}\;\; x^T\Sigma x$$ $$\hspace{15mm}\text{Subject to}\;\; p^Tx = \frac{1}{n}p^T\boldsymbol{1}$$ $$\hspace{25mm}\boldsymbol{1}^Tx=1$$ ...
2
votes
1answer
158 views

Solving the quadratic in the Fast Iterative Method

The Fast Iterative Method is a way of solving the Eikonal Equation on a discrete grid, similar to the Fast Marching Method discussed in another question here. The paper describes the overall ...
1
vote
1answer
410 views

Mixed-integer quadratic programming, state of art [closed]

I used Gurobi with a MIQP with 26 binary variables and 26*4 interaction term without any other constraint. The speed is very slow already.... I want to ask what is the state of art of MIQP solvers. ...
1
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3answers
2k views

Minimize quadratic form with equality constraints

I want to minimize function: $f(x) = x^T \cdot A \cdot x + b \cdot x$ given constraints: $B \cdot x = 0$. Here: $x$ is a vector ($x \in \mathbb{R}^n$), $A$ is a matrix of size $n \times n$, $b$ ...
1
vote
3answers
272 views

rank-deficient NNLS

I want to find the minimum-norm solution to a rank-deficient least-squares problem, subject to positivity constraints, e.g. $$\min_x\ \|x\|^2 \quad s.t.\quad Ax = b,\ x \geq 0$$ where $A$ is large, ...
2
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1answer
393 views

0,1 binary polynomial programming

Is there a mathematical optimization branch that explicitly tries to optimize this (type) problem? $$\eqalign{ & \min \cr & \sum\limits_{i = 1}^N {(J*s[i] + {J_1}*s[i]*s[i + 1] + {J_2}...
1
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1answer
472 views

How can a quadratic positive definite minimization be unbounded [closed]

I am minimising a diagonal quadratic matrix using CPLEX. All off diagonal elements are zero. It has 500 variables and 20 linear constraints plus each variable is constrained to be within 0 and 1 All ...
3
votes
2answers
126 views

A separable nonnegative quadratic program

I have spent quite some time trying to solve the following quadratic program: $$\min \sum_{i=1}^n (\frac{1}{2}x_i^TQx_i+c_i^Tx_i), \quad \mathrm{s.t. } \quad x_i\ge 0 \quad \forall i,$$ where $n$ is ...
0
votes
1answer
672 views

CPLEX claims to have solved QP minimisation but solution is not optimal

I am trying to solve a small QP problem in CPLEX. The problem has several linear constraints. The optimiser runs and finds a solution which satisfies these constraints and CPLEX returns a success code....
3
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1answer
664 views

Sparse quadratic programming solver

For a hobby project I need to solve a series of quadratic programming problems each with about 500 variables about 1000 constraints, each of the form $x_i-x_j\le c_{ij}$ the objective function is the ...
4
votes
3answers
157 views

Plane constraints in R3

I have multiple plane constraints in $\mathbb{R}^3$ of the form: $$n_i \cdot x \ge \delta_i$$ Where $n_i$ is the $i$th plane normal (in form (x, y, z)), $x$ is a point in space, and $\delta_i$ is ...
1
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0answers
238 views

Numerical integral with a weakly singular kernel with a satisfactory precision

I am working on numerical method for time fractional PDE. One problem is that I must compute numerical integral with the following form: \begin{equation} \int_0^{t_m} (t_m-s)^{-\beta}f(s)ds \end{...
3
votes
3answers
612 views

Minimizing a negative definite quadratic function with specified bounds

I have the following quadratic program $$\begin{array}{ll} \text{minimize} & f(x) = \frac{1}{2} x^T D x + c^T x\\ \text{subject to} & x_{\text{lower}} < x < x_{\text{upper}}\end{array}$$...
1
vote
1answer
562 views

Quadratic Programming bound constraints

I have a quadratic programming problem with constraints of the general form: Minimize w.r.t. x: f(x) = (1/2) x^T * Q * x + c^T * x subject to one or more ...
4
votes
1answer
911 views

Solver suggestion for many small quadratic problem in C++

I have a C++ program/model that in some parts already use IPOPT (with ADOL-C and ColPack) to solve some pretty large non linear problems. Now in an other part of the program I need to solve a large ...
1
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0answers
122 views

Quadratic programming problem involving permutation matrices

Does anyone know a good algorithm for quickly finding an approximate solution to the following problem? Given two square matrices $A$ and $B$, minimize $\| P A P^\top - B \|$ over all permutation ...
1
vote
2answers
875 views

Modeling a quadratic constraint with a linear expression

In a problem I am trying to model with a MIP program, the following scenario occurs: I am given binary variables $x_1,\ldots,x_n$ and $y_1,\ldots,y_n$ which can really be regarded as $n$-vectors. ...
1
vote
0answers
97 views

Quadratic optimization without any cross terms

I need to solve a quadratic program in Java which minimizes a sum of squares. The problem is that there are constraints to be satisfied, and so a quadratic optimizer must be used. I've been thinking ...
3
votes
2answers
426 views

Quadratic Programming: Quadprog

Given a real-rectangular matrix $S$ and inorder to solve this simple quadratic programming problem: Minimize $w'S'Sw = ||S w||^2$ over $w$ subject to $e^Tw = 1$ and $w \geq 0$ using a solver I ...