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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4
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0answers
72 views

Quadratic optimisation with quadratic equality constraints

I would like to solve the following optimisation problem: $\min_{x} (x'Ax)$ subject to $x'Bx = x'Cx = 1$. Where A is symmetric and B and C are diagonal. Does anyone have a suggestion for an ...
4
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0answers
145 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 ...
2
votes
0answers
44 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
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0answers
21 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
163 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} & ...
2
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0answers
41 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
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0answers
39 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
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0answers
38 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
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0answers
115 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)= ...
2
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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
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0answers
161 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 ...
1
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0answers
11 views

Quadratic programing problem and MATLAB

I have a little problem with quadratic programing problem: ${\bf v}^T \Sigma {\bf v} \rightarrow min $, and constrains are $ {\bf v}^T \mu = \mu_*, {\bf v}^T {\bf 1 }= 1, 0 \leq {\bf v}. $ Where ...
1
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0answers
18 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} + ...
1
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0answers
33 views

How to prove that this is a Quadratic Optimization Problem?

Show that the following problem is a Quadratic Optimization Problem. $$ \begin{array}{ll} \max & \sum\limits_{i=1}^n \left( \mu_i(z_i + x_i) - a_i \lvert x_i \rvert - b_i x_i^2 \right) \\ ...
1
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0answers
69 views

Linear reformulation or approximation of a quadratic inequality set

Based on the useful comments I reformulated my problem - hopefully it's more clear now. Let $A,B \in \mathbb{R}^{d \times d}$ be symmetric positive-semidefinite matrices, $x \in \mathbb{R}^d$ and ...
1
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0answers
34 views

Projection onto the positive subset of a hyperplane

I am wondering if there is a closed form solution for the projection of a point $x_0$ onto the subset of a hyperplane $Ax=b$ in the positive orthnat, that is $x\geq0$?
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0answers
63 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 ...
1
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0answers
68 views

Quadratic optimization problem (inner products) with stochastic constraints

Let the set of feasible solution be the set of all row-stochastic $n \times k$ matrices $P = [p_{ij}]$, that is $\mathcal{P} := \{P \in \mathbb{R}^{n \times k} \ | \ P \mathbf{1} = \mathbf{1}, P \geq ...
1
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0answers
94 views

Why use two slack variables in the support vector regression formulation?

I am learning support vector regression but cannot fully understand the rational of the slack variable tricks in its formulation. The original optimization problem for SVR is as follows: ...
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0answers
54 views

Two quadratic programming problems always same answer?

Was exploring quadratic programming optimization and for two types of problems the answers seemed to always equal. Is there an intuitive proof? Problem 1: Minimize $\tfrac{1}{2} \mathbf{x}^T ...
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0answers
46 views

How to interpolate a function with a reproducing kernel

I am trying to interpolate a function that is noisy, but I know with a high amount of certainty about a third of the points in the series. I am trying to estimate the smooth mean of the signal via a ...
1
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0answers
94 views

Modeling a lower-bound constraint on a euclidean distance in quadratic programming

I have been trying to model a certain problem into a mainly linear program, but with some quadratic constraints since I don't think that is something that can be avoided. I hope to solve it either ...
1
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0answers
14 views

Optimizing a set of rules to better predict the outcome of events

I'm trying to better predict the top three finishers of the next 1000 800m mens freestyle swimming race. I've got a set of rules to rate the swimmers: 1) Add 5 points if the swimmer won his last ...
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0answers
39 views

Reduce degree of a high degree unconstrained binary term to quadratic unconstrained binary term

I'm working on a optimization project, in this project I have to convert higher order unconstrained binary polynomial to quadratic unconstrained binary polynomial. Can anyone give me a hint of how to ...
1
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0answers
71 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} ...
1
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0answers
46 views

Existence criterion for solution in quadratic programming

I have the problem $$ \begin{align*}\min \quad&f(x)= c^Tx + x^TQx \\ &x\in D \end{align*}$$ with $D=\{ x \in \mathbb{R}^n \mid Ax \leq b\}$, $A,Q\in \mathbb{R}^{n\times n}$ and $b,c \in ...
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0answers
144 views

Determining initial values for optimization problem

I am trying to solve an optimization problem with a quadratic objective function and non-linear constraints, using SQP (Sequential Quadratic Programming). I am attempting at doing the implementation ...
1
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0answers
47 views

Hinge point in quadratic program (bilateral constraint)

My question itself is possibly quite simple and I guess that if someone can answer me they probably does not need a wall of text that is my background to the problem, but I figured I should provide as ...
1
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0answers
22 views

Uniqueness of the solution to a quadratic opt problem

Consider a positive definite matrix $\boldsymbol H$, the known vectors ${\boldsymbol b}$ and ${\boldsymbol a}_i$. Now the minimization problem is casted with respect to the vector ${\boldsymbol x} $ ...
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0answers
73 views

Linear Complementarity Problem - multiple solutions, which one will it find?

If I have a inequality constrained system: w = Mz + q <= 0, z<=0, z^T w = 0 that for some given properties M and ...
1
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0answers
157 views

Minimize a complex quadratic subject to two convex quadratic constraints

I have the following the optimization problem \begin{align} \min_{\mathbb{x}\in \mathbb{C}^{N \times 1}}~&||\mathbb{x}^H\mathbb{u}||_2^2-2*Real\{ \mathbb{x^Hu}\} \\\ ...
1
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0answers
202 views

Quadratic programming with simplex constraints

I have this quadratic function $x'Ax$ which is not convex. I want to maximize this function subject to the constraints that the solution x lies in a simplex such that $\sum_{i=1}^{n}x_i=1$. That means ...
1
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0answers
138 views

Algorithm and solver for large, dense, positive-semidefinite integer QP

I am interested in the solutions of a very large quadratic programming (QP) problem \begin{align} \min_{x \in \mathbb{R}^n} & x^T Q x\\ \mathrm{subject\ to} & A x = b\\ & x \in \{0,1\}^n ...
0
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0answers
36 views

Solution of quadratic optimization with linear constraints

Hi, I want to solve a quadratic optimization problem defined as $\min \|\bf{a^T}\bf{x}\|^2_2$ $s.t. \|\bf{x}\|_1=1$ $\ \ \ \ \ \ \ \ x_i\ge0,\ i=0,1,...,n-1$ $\ \ \ \ \ \ \ \ \bf{b}^T\bf{x}-C\le0$ ...
0
votes
0answers
42 views

Quadratic optimization: Why does the formula have “$\frac{1}{2}”$ in front?

$$\frac{1}{2}x^THx+c^Tx + c_0$$ I have just formulated a problem as a quadratic optimization problem in two variables. My solution differs from the solution manual in the aspect that they have, only ...
0
votes
0answers
19 views

(Convex) reformulation of a nonlinear program

Consider the following program: \begin{eqnarray*} \min_{\mathrm x}\sum_{i=1}^{n}{\sum_{j=1}^{n}{\big(x_i(Sx)_i-x_j(Sx)_j\big)^2}}\\ \mathrm{subject\; to}\quad \sum_{i=1}^{n}{x_i}=1 \\ x_i\geq 0 ...
0
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0answers
18 views

Minimize/reformulate sum of products as convex problem

Is it possible to optimize this objective function as such or transform it into an convex formulation? The unknown continuous variables $x_i \in [-1, 1]$ are nodes in a graph and for each edge ...
0
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0answers
9 views

Nonlinear Programming

I have the following non-linear programmig problem that I have arrived at after various manipulations. I have to find the set of values for $x$ and $y$ that satisfy the following: $$ x^{n}+y^{m}=C $$ ...
0
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0answers
46 views

Quadratic programming over nonnegative orthant?

I'm trying to solve a quadratic optimization over nonnegative orthant as below. \begin{equation} \begin{aligned} \mbox{maximize} \quad & -\frac{1}{2} \lambda^{T} [\frac{\Sigma_{i}}{\alpha_{i}} + ...
0
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0answers
24 views

convex symmetric quadratic programming

Good morning, I have trouble understanding the concept of self-duality in quadratic programming. I am reading a paper right now where we first show that the dual of a convex symmetric quadratic ...
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0answers
31 views

Can variance be replaced by absolute value in this objective function?

Initially I modeled my objective function as follows: argmin var(f(x),g(x))+var(c(x),d(x)) where f,g,c,d are linear functions in order to be able to use mixed integer linear solvers, I modeled the ...
0
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0answers
39 views

Quadratic Programming “big M” method

How does the optimization problem $$\min_{x,\eta} \frac{1}{2} x^TGx+x^Tc+M\eta$$ $$ s.t. Ax+\eta-b \geq0$$ $$\eta\geq0$$ look in standard form? What would the KKT look like? The problem is, that ...
0
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0answers
28 views

Quadratic optimization problem with quadratic equality constraint

I am trying to solve the following optimization problem: $$ \min_{x \in \mathbf{R}^2} \, x^T A x + b^Tx \quad \text{subject to $x^T J x = 1$} $$ where $A$ is a positive semi-definite $2 \times 2$ ...
0
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0answers
33 views

How to do linear quadratic dynamic programming with non homogeneous quadratic equation

I am not well versed on matrix algebra and linear quadratic programming. I am wondering if it is possible to make a non-homogeneous equation into a homogeneous one. I need to make the following ...
0
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0answers
70 views

Computational complexity of the following quadratic program (QP)

Let $A^TA$ be a $n \times n$ matrix. I have the following quadratic program to solve: \begin{array}{rl} \min \limits_{x} & x^T A^T A x \\ \mbox{subject to} & \sum_{i=1}^{r} x_i =1, ...
0
votes
0answers
20 views

Existence of lagrange multipliers with polyhedral constraints

I am working with a paper (Exact regularization of polyhedral norms, Schöpfer 2012) which states as a well-known fact that, if $f$ is a polyhedral norm, then for some $\mu^* > 0$ \begin{equation} ...
0
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0answers
38 views

Complexity of solving quadratic programming (QP) using ADMM

I am trying to analysis the time complexity of quadratic programming problem, using alternating direction method of multipliers. Anyone could give me a help?
0
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0answers
45 views

Approximate inverse (or fast optimization) of non-linear least squares problem

Problem Statement Let ${\bf x}\in\mathbb{R}^N$ and ${\bf W}\in\mathbb{R}^{K\times N}$, ${\bf V}\in\mathbb{R}^{N\times K}$. We define $${\bf y} = f({\bf x}) = [{\bf V}[{\bf Wx}]_+]_+$$ where $[.]_+ = ...
0
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0answers
51 views

How to obtain the optimal lagrange multiplier vectors if the globally optimal solution for a nonconvex QCQP is found.?

I am using a blackbox solver to solve the following nonconvex QCQP to global optimality. $$ \min_x x^TQ_0x + c^T x \\ s.t. \quad x^TQ_1x+c_1^Tx=b_1 \\ Ax=b \\ l\leq x\leq u $$ where $Q_0$ is ...
0
votes
0answers
47 views

When is a quadratically constrained quadratic program (indefinite objective matrix) unbounded?

I have a nonconvex QCQP of the form $$x^TQ_0x + c^T x$$ such that $x^TQ_1x+c_1^Tx=b_1$, $Ax=b$, and $l\leq x\leq m$ where $Q_0$ is indefinite diagonal matrix and $Q_1$ is positive semidefinite ...