# 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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I would like to solve the following trace-minimization under equality constraints optimization problem: $$W^* =\arg\min \operatorname{Tr}[WCW^T] \text{ s.t. } A=B^TW^TWB$$ where $W,C\in\mathbb{R}^{... 0answers 10 views ### Lagrangian Relaxation of quadratically constrained quadratic program I have the following problem: $$\min_{w,\theta\ge0}\frac{1}{2}\|w-w_t\|^2+(\theta-\theta_t)^2 \text{ s.t. } w^\top(\hat n\hat z-nz)+\theta w^\top(z-\hat z)+1 \le 0,\theta-1\le 0$$ Notice that$w$is ... 0answers 43 views ### Quadratic problem with two vectors linked by one quadratic constraint I would like to find $$\min_{w,b} w_iA_{ij}w_{j} + b_iB_{ij}b_j + 2\alpha_iw_i + 2\beta_ib_i$$ Constrained to: $$w_i > 0$$ $$\sum w_i = 1$$ $$b_i=w_i^2$$ Where$A$and$B$are positive ... 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\...
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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 ...
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### 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 $c_S \in \mathbb{R}$,...
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### 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{...
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### 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 ...
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The problem that I am going to describe arises from compressed sensing technique and after using weighted least squares it can be transformed into the following least squares problem: \underset{\... 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 46 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 20 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 68 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 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 ... 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}... 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$$It says that it have to equal an ellipse with center at the point (2000,3000) and a horizontal ... 0answers 26 views ### Nonlinear Optimization I have a nonlinear optimization problem, but constraints are ODE. Cost function is J= x1+x1*x2+x1^2 while constraints are, \underline{x_i} < x < \bar{x_i} (for i=1,2,3) ; \frac{dx3}{dt}=... 1answer 30 views ### Quadratic programing problem and MATLAB I have a little problem with quadratic programing problem: {\bf v}^T \Sigma {\bf v} \rightarrow min , and constrains are  {\bf v}^T \mu = \mu_*, {\bf v}^T {\bf 1 }= 1, 0 \leq {\bf v}.  Where ... 1answer 58 views ### Solve \min_{\mathbf{x}} \sum_i \min\left[ (\mathbf{c}_i^T\mathbf{x}-a_i)^2, (\mathbf{d}_i^T\mathbf{x}-b_i)^2 \right] I am wondering if there is an efficient (perhaps closed form) way to solve the following piecewise quadratic minimisation problem:$$ \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 ...