This is for questions on Quadratic Programming (QP). A QP problem is the problem of minimising or maximising a quadratic objective function subject to affine constraints.

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6
votes
2answers
730 views

What numerical methods are known to solve $L_1$ regularized quadratic programming problems?

What numerical methods are suitable to solve the following problem $$\min_x \tfrac{1}{2}x^T A x + b^Tx + \lambda ||x||_1$$ where $x,b\in\mathbf{R}^n$, and $A\in \mathbf{R}^{n\times n}$ is positive ...
6
votes
3answers
85 views

Is a sinc-distance matrix positive semidefinite?

I've been trying to crack this problem for days but I can't find a way around it. Given a set of unique N points $X = {x_1,..x,N}, x_i \in R^3$, the associated sinc-distance matrix $S \in R^{n\times n}...
6
votes
1answer
647 views

Linear Programming with One Quadratic Equality Constraint

I have a problem which can be formulated as a Linear Programming with One Quadratic Equality Constraint: where variable x is n-dimensional vector and H is a Semi-Positive Definite n-by-n matrix. I ...
5
votes
2answers
216 views

solution to $\min \|A-BXC \|$

I have the following problem. Let $A$, $B$ $C$ be real-valued matrices of size $m \times q$, $m \times n$, $p\times q$ respectively. I would like to find matrix $X$ of size $n\times p$ and maximum ...
5
votes
0answers
91 views

Quadratic optimisation with quadratic equality constraints

I would like to solve the following optimisation problem: $$\text{minimize} \quad x'Ax \qquad \qquad \text{subject to} \quad x'Bx = x'Cx = 1$$ Where $A$ is symmetric and $B$ and $C$ are diagonal. ...
4
votes
2answers
212 views

Duality in quadratically constrained quadratic program

I have been given the primal quadratic program with a single quadratic constraint as given below: $$ \text{min} ~~~~~~~~~~~~~~~~~~~~~~~~~ \frac{1}{2}x^{T}Qx $$ \begin{align*} \text{subject to}~~~~~~~~...
4
votes
1answer
714 views

How to use lagrange multipliers here?

I have a simple QP as below: $\min L(x,y) = (x-5.1)^2+y^2$ such that $(x-3)^2+y^2\geq1$ $(x-5.3)^2+y^2\geq1$ $(x-7)^2+y^2\geq1$ Intuitively, I think the optimal solution of the problem is $x^*=4....
4
votes
1answer
111 views

Find closest point, subject to linear inequality constraints

Given a point $p\in \mathcal{R}^2$, I want to compute the closest point $x \in \mathcal{R}^2$, subject to linear inequality constraints $Ax \leq b$. That is, $$\begin{array}{ll} \text{minimize} & ...
4
votes
2answers
814 views

Analog of Simplex Method for Quadratic Programming

It happened so I need to work with an algorithm for solving QP problems. The main issue here is that I can't find any references to this algorithms (at least no references in sources in english). ...
4
votes
2answers
2k views

what is the computational complexity of solving a quadratic program with linear inequality constraints

I'm aware of several solution methods and have several solvers at my disposal, but I can't for the life of me find analysis on the complexity. In particular, I'm interested in the complexity of ...
4
votes
0answers
201 views

How to find accuracy of Matlab's quadprog solver?

I have solved with quadprog from Matlab a strong convex quadratic problem given as $$ f(x) = x^TQx + c^Tx$$ with constrains $$ Cx \leq b.$$ Now the output of quadprog is: Minimum found that ...
3
votes
1answer
1k views

Are “constrained linear least squares” and “quadratic programming” the same thing?

A Quadratic Programming problem is to minimize: $f(\mathbf{x}) = \tfrac{1}{2} \mathbf{x}^T Q\mathbf{x} + \mathbf{c}^T \mathbf{x}$ subject to $A\mathbf{x} \leq \mathbf b$; $C\mathbf{x} = \mathbf d$; ...
3
votes
2answers
86 views

$ k x^2 +4x = n $, Algorithm or any other method needed

I want to find any $n < 10^{18} $ so that the equation below has at least two pairs of solutions $(k, x)$ $ k x^2 +4 x = n $ constraints: $x > 10^6; \; x > k ; \; k, x \in \mathbb{N}$ I ...
3
votes
2answers
566 views

A standard quadratic minimization problem

Consider the "Complex" Quadratic minimization problem \begin{align} \min_{\mathbb{x}\in \mathbb{C}^{N \times 1}}~\mathbf{{x}}^H\mathbf{Q}\mathbf{x}-2~\Re{(\mathbf{x}^H\mathbf{b})}+1 \end{align} $\Re(...
3
votes
2answers
403 views

Quadratic equation to calculate a temperature from resistance

I'm trying to implement an electronic temperature sensor that gives a resistance value. The sensor is a Honeywell TD4. In the datasheet, they give a table of values : -40ºC => 1584Ω ±12Ω -30ºC => ...
3
votes
1answer
48 views

How to solve the following quadratic matrix equation?

Solve the following matrix equation in $D$ $$ A=D^{T}(DVD^{T}+\alpha\lambda_{\max}(D^{T}D)I)^{-1}D$$ where $I$ is the identity matrix, $A$ and $V$ are known matrices, $\alpha$ is a known ...
3
votes
1answer
207 views

binary quadratic optimization problem

I am trying to solve the following binary quadratic program. $$ \min_{\Delta} \Delta^T H \Delta + c^T\Delta \\ \text{Such that:} ~~~\Delta\in \{0,1\}^n ~~\text{and}~~ \sum_{i=1}^n \Delta_i \leq \Gamma ...
3
votes
2answers
900 views

Quadratic Equality Constraints via SDP

I want to know if it is possible to solve a QCQP problem with quadratic equality constraints in SDP. I know it is possible to convert a QCQP to an SDP by using the Shur complement. The following ...
3
votes
1answer
454 views

Shortest Path and Minimum Curvature Path - implementation

Let's say we are given a race track, which may be described as a closed curve of given width (it may differ along the curve). My task is to implement an algorithm which finds two kinds of trajectories ...
3
votes
1answer
2k views

How to convert quadratic programming problem to matrix form

I am new to this topic and am looking at an example I can't figure out. Can someone please help explain how this example creates the matrices used in the solver? Thanks! This is the PROBLEM ...
3
votes
1answer
83 views

Finiteness of the Supremum of Inner Product of Two Finite Sum Positive Sequence

Let $$A = \Big\{(a_1,a_2,\dots)\ \Big|\ a_i\ge 0, \sum_{i=1}^\infty a_i=1\Big\},$$ $$v(x)=\sup\left(\bigg\{\sum_{i=1}^\infty a_ib_i\ \bigg|\ (a_i)_{i=1}^\infty,\, (b_i)_{i=1}^\infty \in A,\,\sup\...
3
votes
1answer
244 views

Optimization problem with ratio objective

I need to solve the following optimization problem $$ \text{maximize} \quad \frac{(a^T x)^2}{x^TBx+c^T|x|} \quad \text{subject to} \quad \|x\|_1=1 \quad (\text{or alternatively} \quad c^T|x|=1), $$ ...
2
votes
2answers
41 views

Maximizing a convex quadratic function in CVX and Matlab

I understand that a convex function can not be maximized as there is no such value. However, consider the following function: $$\begin{array}{ll} \text{maximize} & 3x^2 + 5y^2\\ \text{subject to} ...
2
votes
1answer
89 views

Find my mistake: Solving quadratic programm by “brute force” triangularization

Do you know the feeling when you have proven something totally incredible? In fact, it is so incredible you do not believe it yourself. So you start looking for your mistake, for the one, small ...
2
votes
2answers
252 views

Finding the closest vector subject to an absolute constraint

I'm trying to solve the following problem: $$\min_b \|d-b\| \\ \text{s.t. } |Ab|^2 \leq y $$ or equivalently $$\min_b \|d-b\| \\ \text{s.t. } |Ab| \leq c = \sqrt{y} $$ Both $d$ and $b$ are vectors. I ...
2
votes
1answer
38 views

Minimizing the Frobenius norm with linear inequality constraints

How to solve the following system for $\mathbf{C}$ and $\mathbf{a}$: $\min\|\mathbf{X-XC} \,\mbox{diag} (\mathbf{a})\|_F^2$ subject to $\mathbf{c}_{ik}\geq 0$, $1^T \mathbf{c}_k = 1$ and $1-\delta\...
2
votes
1answer
35 views

Solving a quadratic convex optimization problem

There's this convex optimization problem which I got stuck after writing the Lagrange equation. I simply couldn't find a way to eliminate the Lagrange multiplier. $$\begin{array}{ll} \text{minimize} &...
2
votes
1answer
66 views

Maximize sum of N variables squared subject to constraint

I have the following problem. Maximize ${\sum_{x=0}^N \beta_{1,x}v_x + \beta_{2,x}v_x^2}$ subject to $\sum_{x=0}^N v_x = X$ wrt $v_x$. I am thinking about both Lagrange multipliers and quadratic ...
2
votes
1answer
43 views

What is the solution for this quadratic program?

Given scalars $p_1\geq p_2\geq \cdots \geq p_r > 0$, can we find a solution for following problem? \begin{align} \text{minimize} & & & \sum_{j=1}^{r} p_j (1-t_j)^2 \\ \text{s.t.} \\ &...
2
votes
0answers
41 views

Boolean Quadratic Programming

I am new to optimization and I am trying to understand concepts of semi-definite relaxation (SDR) through examples. It seems my understanding of this topic is not fully clear as I will show in details ...
2
votes
0answers
26 views

How to efficiently solve a quadratic program repeatedly?

I have a quadratic problem, \begin{align} \min_{x\in [0,1]^n} x^T p+ \frac{1}{2\lambda} x^T Q x \end{align} ($Q$ is semidefinite) which I want to solve repeatedly, with the slight change of p and Q, ...
2
votes
0answers
29 views

Theoretically understanding the algorithmic solution to this Quadratically Constrained Quadratic Optimization Problem

I have a simple QCQP problem to solve: $\min_{t} x(t)^{T}Ax(t)$ subject to constraints $x(t)^{T}Ax(t) > 1 $ where A is a positive definite matrix and $x(t) \in \mathbb{R}^2$ is some time ...
2
votes
0answers
25 views

Transformation into quadratic program

I am struggling with the following optimization problem: \begin{equation*} \begin{aligned} & \underset{x_{1}, \ ..., \ x_{5} \in \mathrm{R}}{\text{minimize}} & & \begin{pmatrix} x_{1} + ...
2
votes
0answers
47 views

Reformulate absolute value as quadratic problem

I'm looking for standard approach to reformulate this objective function. The aim is to find values of $x_i$ that are close to either $y_i$ or $-y_i$ ($y_i$s are known) in a least-squares sense: $...
2
votes
0answers
109 views

Relaxation of non-convex QCQP with one quadratic and one linear constraint

According to Boyd we know that a non-convex QCQP problem with one quadratic constraint has strong duality with the relaxed SDP or Lagrange counterpart. (check "Convex Optimization" by Boyd, Appendix B)...
2
votes
0answers
24 views

Quadratic programming

For a research project I am interested in minimizing $x^TQx$ under the constraints $Qx \le b$, where $Q$ is a positive definite matrix and $b$ is a vector of negative elements. I have solved this ...
2
votes
0answers
165 views

Lagrange multiplier expression

I would like to solve the following optimization problem using the gradient ascend method: \begin{array}{ll} \text{maximize}_{\theta} & \theta^TQ_1\theta + b_1\\ \text{subject to} & \theta^...
2
votes
0answers
48 views

Nearest non-negative solution for $Av=b$

Let $A$ be a $n\times m$ matrix. Let us define the system $$Av=b$$ $$v\geq 0$$ I want to find a solution $v$ of this system that is the closest (euclidean norm) to $v_0$, a given $n$-dimensional ...
2
votes
0answers
41 views

Effective convexity criterion for the finite point set in $\mathbb{R}^3$

I need to find effective convexity criterion for the finite point set. Below there is description of what is meant by "effective" criterion. Definition. Let $M = \{A_{1}, \ldots, A_{n}\}$ be the ...
2
votes
0answers
78 views

Least squares with three quadratic constraints (Ellipse fitting based on algebraic distance)

I would like to fit an ellipse to a given set of scattered data in $\mathcal{R}^2$. The fitting problem is in form least squares, minimizing the sum of squared algebraic distances \begin{equation} \...
2
votes
0answers
44 views

Reproducing Kernel Function Interpolation

My problem is as follows: I am attempting to use a reproducing kernel and quadratic programming to optimize/interpolate a smooth function from constraints. I am using all local extrema to calculate ...
2
votes
0answers
160 views

Constrained Quadratic Optimization(Reproducing Kernel)

I am attempting to use a constrained quadratic optimization to find the coefficients of a reproducing kernel. The problem is as follows: $y(t)=\sum_{i=0}^J\alpha_iK(t, t_i)$ $Q(\alpha)= \alpha^tK\...
2
votes
0answers
58 views

Quadratic Integer Programming

Would anyone mind helping me solve this problem $$ \min\space f(x) = \frac12 x^\mathrm TQx + bx + c \qquad \text{s.t. } \sum_i x_i=\lambda $$ where $x$ is a vector whose entries are positive ...
2
votes
0answers
180 views

Nearest point to a convex polytope

I am looking for fast, memory-efficient computational algorithms to solve the following problem: Minimize: $||x - x*||_2^2$, subject to constraints $A x = a, B x <= b, l <= x <= u$, where $...
2
votes
1answer
311 views

Convex optimization and linear programming please help! :)

How would I write the following as a standard form LP? Minimizing $\sum_{i=1}^n x_i + c\max(a_i-x_i)$ for $a_i \ge 0$ and what is the optimal value for when $c=n$ How to express minimize $\frac{1}{2} |...
2
votes
1answer
128 views

Software to optimize a quadratic program with quadratic constraints

I'm working in eight dimensions and want to minimize $x^TAx$ under the constraints $x^TBx \geq c$. Unfortunately, A is not positive semidefinite. Worse, I am almost positive that my domain is not ...
2
votes
2answers
67 views

How do I solve the following equality-constrained quadratic program?

I am trying to minimize: $$(x_1-k_1)^2 + (x_2-k_2)^2 + (x_3-k_3)^2 +\ldots+ (x_n-k_n)^2$$ subject to following equality: $$B = 1 + x_1 + x_2 + x_3 + x_4+\ldots+x_n.$$ Is there a closed form ...
1
vote
4answers
102 views

Solution to a system of quadratics

I am learning about a Bell State, and am trying to show that they are entangled. I believe that the required proof is to show that the system $$\alpha_0^2+\alpha_1^2=1$$ $$\beta_0^2+\beta_1^2=1$$ $$\...
1
vote
2answers
70 views

Quadratic programming for special equation issues

My problem is how to find $\tau_1$ and $\tau_2$ s.t maximize the objective function is $$E=M-\alpha V$$ subject to $$-0.0062\le\tau_1\le0.499$$ $$-0.479\le\tau_2\le0.0262$$ $$\tau_1+\tau_2\le0.02$$ ...
1
vote
4answers
89 views

Optimization of a quadratic function with qudratic constraints

I'm a Graduate student of Electrical Engineering. I have some basic knowledge on Convex Optimization. For my research, I cam across the following optimization program. With $\mu > 0$, find $\arg \...