# Tagged Questions

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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### Given a set of vectors and a target vector, find the set of scaling factors that minimizes distance of sum of those vectors from target

I have a set of $n$ starting vectors $\vec i_n$ and a target vector $\vec t$. I have a set of scaling factors $a_n$ for which I can compute the sum $\vec s$: $$\vec s = \sum_{i=1}^n {a_i \vec i_i}$$...
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### 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 ...
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### How to calculate a projection matrix for nonnegative constrained least squares?

Suppose we have a data vector $\boldsymbol{z}$ in R^{p} and a training data matrix $\boldsymbol{X}$ in $R^{p \times N}$, where N (N>p) is the number of samples in the training data matrix. If we'd ...
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### QP with simplex constraints. [closed]

I want to solve something: $\Vert \mathbf{x-Da}\Vert_2^2$ s.t. $\sum \mathbf{a}_i=1$ and $\mathbf{a}_i\geq 0$. How to convert such problems into equivelent Quadratic program so that I can use Matlab'...
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### Converting a sum over squared norms to Quadratic Programming problem

I'm trying to minimize the following function which is a function of a set of vectors $\vec{x}_i \in \mathbb{R}^2$, and is parametrized by the fixed parameters $\alpha, \beta,\gamma_i,\vec{u}_i$ where ...
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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. $$\begin{array}{ll} \text{minimize} &... 1answer 15 views ### Tree-width of a quadratic pseudo-Boolean function A pseudo-Boolean function f : \mathbb{B}^n \mapsto \mathbb{R} is of the following form.$$ f \left(x_1, \ldots, x_n\right) = \sum_{S\subseteq V} c_S \prod_{j \in S} x_j Here c_S \in \mathbb{R},... 0answers 33 views ### Residual norm of active set method in non negative least squares I am having trouble with understanding one of the statements in active set methods done for non negative least squares given in Book written by Lawson. The quadratic problem can be written as \begin{... 0answers 21 views ### Quadratization of 5 x_1 x_2 − 7 x_1 x_2 x_3 x_4 + 2 x_1 x_2 x_3 x_5 using Rosenberg's algorithm In section 4.4 of Pseudo-Boolean Optimization by Boros et al., the authors have reproduced Rosenberg's quadratization algorithm as follows. Then they have given an example of implementing the ... 0answers 10 views ### Bipartite Matching with quadratic objective I'm looking for the best way to formulate and solve the following bipartite matching problem: I have n nodes on the left hand side of the diagram, partitioned into ... 1answer 43 views ### Convex optimization: Piece-wise, quadratic objective This question is about convex optimization with a convex objective function, which is defined piecewise. We have two functions, a concave function A(x) and a strictly convex, increasing function B(x), ... 0answers 13 views ### Matrix inversion vs. iterative scheme in quadratic programming problem for high dimensions According to this entry about quadratic programming problems with equality constraints the system will not be positive definite any longer for higher dimensions. For lower dimensions this system could ... 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, ... 0answers 21 views ### Finding a suitable solver I have a problem finding a solver that can solve a mathematical programming model with a quasi quadratic object function. I have tried some commercially available quadratic and non-linear solvers, but ... 0answers 19 views ### Binding inequality constraints in linear programming with quadratic constraints I am trying to maximize the following objective function: a_{1}b_{1}x_{1}+a_{2}b_{2}x_{2}+a_{3}b_{3}x_{3}+a_{4}b_{4}x_{4} The quadratic constraint is given by b_{1}^2 x_{1}^2 + b_{2}^2 x_{2}^2 + ... 1answer 67 views ### A light solution of a quadratic programming problem I have a simple and light quadratic programming problem that I need to solve, as following: \begin{align} & \underset{x}{\arg\min} & & \dfrac{1}{2}x^T x-z^T x\\ & \text{subject ... 0answers 28 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 ... 2answers 4k views ### Using Matlab quadprog to solve markowitz model I have the markowitz model shown below and I need to use the quadprog function to solve it (i.e get the values for w_i values). However I am a bit new to mat lab and not sure which definition of ... 1answer 20 views ### Why in quadratic programming is particularly simple when there are only equality constraints; specifically, the problem is linear I found in wikipedia that... Quadratic programming is particularly simple when there are only equality constraints; specifically, the problem is linear font: Quadratic Programming What I can'... 1answer 22 views ### General form to standard form regarding ellipse? I've tried 2 hours to do this so I hope someone can help me:11400000=-0.64x^2+2560x-y^2+6000y$$It says that it have to equal an ellipse with center at the point (2000,3000) and a horizontal ... 1answer 30 views ### Quadratic programing problem and MATLAB I have a little problem with quadratic programing problem: {\bf v}^T \Sigma {\bf v} \rightarrow min , and constrains are  {\bf v}^T \mu = \mu_*, {\bf v}^T {\bf 1 }= 1, 0 \leq {\bf v}.  Where ... 0answers 25 views ### Nonlinear Optimization I have a nonlinear optimization problem, but constraints are ODE. Cost function is J= x1+x1*x2+x1^2 while constraints are, \underline{x_i} < x < \bar{x_i} (for i=1,2,3) ; \frac{dx3}{dt}=... 1answer 58 views ### Solve \min_{\mathbf{x}} \sum_i \min\left[ (\mathbf{c}_i^T\mathbf{x}-a_i)^2, (\mathbf{d}_i^T\mathbf{x}-b_i)^2 \right] I am wondering if there is an efficient (perhaps closed form) way to solve the following piecewise quadratic minimisation problem:$$ \min_{\mathbf{x}} \sum_{i=1}^n \min\left[ (\mathbf{c}_i^T\mathbf{x}...
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Let $f$ be a real valued quadratic function of 2 real variables: $$f(x,y) = ax^2 + by^2 + cxy + dx + ey + f$$ How to minimize it? Subject to constraints: $$0\leq x \leq 1, \quad 0\leq y \leq 1$$ ...
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### Solution of quadratic optimization with linear constraints

Hi， I want to solve a quadratic optimization problem defined as $\min \|\bf{a^T}\bf{x}\|^2_2$ $s.t. \|\bf{x}\|_1=1$ $\ \ \ \ \ \ \ \ x_i\ge0,\ i=0,1,...,n-1$ $\ \ \ \ \ \ \ \ \bf{b}^T\bf{x}-C\le0$ ...
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There are a few questions on this topic already. However, none of them really answer my question. The most relevant are these: Quadratic optimisation with quadratic equality constraints Quadratic ...
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