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Questions tagged [quadratic-programming]

Questions on quadratic programming, the optimization of a quadratic objective function subject to affine constraints.

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1answer
20 views

quadratic programming /symmetric matrix

I have a quadratic program with $ F: \mathbb{R^n} \rightarrow \mathbb{R}, F(x)=x^TQx$ I want to find a symmetric matrix M for Q, such that $F(x)=x^TMx$ holds for all x. I can write Q as sum of ...
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2answers
71 views

Determining the coefficients of $y = a x^2 + b x + c$ so that the graph contains three given points

Determine coefficients of the equation for a second order polynomial in the format: $y = a x^2 + b x + c$ Here are the $x$-$y$ coordinates of three points: $(0.8143,0.3500)$ $(0.2435,0.1966)$ $(0....
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1answer
46 views

Unconstrained convex quadratic integer programing

Let $N_i=\{0,1,\dots,\bar{n}_i\}$ and define $N=N_1\times \dots \times N_I$. I have a minimization problem of the form $$ \min_{n\in N} \sum_i A_i n_i +\sum_i \sum_{j\neq i} B_{ij} (n_i-n_j)^2 $$ ...
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1answer
37 views

Approximate function by stacking building blocks

I need some help with a 'generalised Lego problem': Given a function $f(x)\geq 0$ on an interval $[a,b]$, and a 'building block' function $p(x)\geq 0$ with compact support. The maximum of f shall be ...
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1answer
34 views

Linearly constrained quadratic program

I have the following quadratic program $$\min_x x^TAx \qquad \text{s.t} \quad Ax \in [a,b]^m$$ where matrix $A$ is positive semidefinite, and is similar both the objective function and in the ...
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0answers
19 views

Is there a time polynomial algorithm to solve a box-constrained quadratic convex program?

I am working on the following quadratic convex problem $min_{x}x^{T}Qx+f^{T}x$ subject to $a\leq x\leq b$ where $x,a,b,f\in\mathbb{R}^{n}$ and $Q$ is positive definite. Is there any algorithm that ...
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1answer
40 views

Measuring infeasibility in convex optimization, relations with dual problem

A question regarding convex optimization and (maybe) duality. I have a problem in the form \begin{align} x^* = \mathrm{arg} \min_x f(x) \quad \text{s.t.} \quad A x \leq b, C x = d, \end{align} where $...
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0answers
22 views

Derive LCP from KKT conditions of a QP

I'm working through this tutorial on LCPs and interior point methods. In it, the authors claim that the following quadratic program $$ \begin{aligned} \min \quad& \frac{1}{2}u^TQu - c^Tu\\ \text{...
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1answer
20 views

Is there only one unique minimizer for this scalar complex quadratic form?

Consider $c,b \in \mathbb{C}$ and $f: \mathbb{C} \mapsto \mathbb{R}$, $$f(c) = bc' + b'c + cc'$$ is there only one extrema of $f$ corresponds to $$c^* = -b$$? where, $'$ means complex conjugate. ...
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1answer
35 views

Local optimality of non-convex quadratic minimization with linear constraints

We are interested in minimizing a quadratic function, which is not convex. The feasible set is a polyhedron. We know that the global minimum is at a vertex when the function is concave. Also, there ...
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0answers
39 views

Quadratic program reformulation maximum to minimum

I am newbie in optimization problem, I have the following optimization problem: $$\max \quad \frac{1}{2}x^THx - q^THx$$ $$\text{s.t.} \;\;l\leq x \leq u$$ where $H, q$ are known constants, and H is ...
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2answers
47 views

Quadratic optimization problem with quadratic constraint

Given $A = A^T \in \mathbb{R}^{n \times n}, X = \begin{bmatrix}x & \eta\end{bmatrix} \text{with} \ x, \eta \in \mathbb{R}^n$ and $||x|| = ||\eta|| = 1$ (vectors in $\mathbb{R}^n$ are treated as ...
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0answers
42 views

Karmarkar's canonical form for convex quadratic programs??

I'm working through this paper A Simple Polynomial-Time Algorithm for Convex Quadratic Programming which gives an interior point algorithm for convex quadratic programming. I've reached the end of the ...
0
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1answer
21 views

confusion in proof of lemma for LCP

I'm looking at the proof of lemma 2 on page 6 of https://cowles.yale.edu/sites/default/files/files/pub/d05/d0549.pdf Which claims that the LCP $$ Mx+q \geq 0 \\ x \geq 0 \\ x^{T}(Mx+q) = 0 $$ has a ...
0
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1answer
53 views

minimum of $\frac{1}{2}x^tQx-c^tx$ subject to $Ax=b$ is local $\iff$ is global

$$\min \frac{1}{2}x^tQx-c^tx\\ Ax=b$$ where $Q\in \mathbb{R}^{n\times n}$ is symmetric, $c\in \mathbb{R}^n$, $A\in \mathbb{R}^{m\times n}$ and $b\in\mathbb{R}^m$. Prove that if ${x}^*$ is a local ...
3
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2answers
168 views

How to minimize $\Vert Ax \Vert$ subject to $\hat{b}^Tx=0$ and $\Vert x \Vert = 1$?

Given $n\times 3$ matrix $A$ and unit vector $\hat{b}\in R^3$, is there a closed-form solution for $x\in R^3$ that minimizes the Euclidean norm $\Vert Ax \Vert$ subject to $\hat{b}^Tx=0$ and $\Vert x \...
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0answers
10 views

Two dimensional sorting: finding $\max_\sigma \langle M^\sigma,C\rangle$ for permutation function $\sigma$

Suppose that we have a non-negative matrices $C,M\in \mathbb{R}^{m\times m}_{\ge0}$. Also suppose $\sigma:[m] \rightarrow [m]$ is a permutation of $\{1,2,...,m\}$. We want to find the permutation that ...
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0answers
48 views

Large scale mixed integer (quadratic) programming

Here we have this optimization problem: Given positive semi-definite matrix $A \in \mathbb{R}^{n \times n }$, and matrix $B \in \mathbb{R}^{n \times m} \text{ and vectors } d \in \mathbb{R}^n,e \in ...
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2answers
140 views

Transform an optimisation problem into a linearly-constrained quadratic program?

I would like your help with a minimisation problem. The minimisation problem would be a linearly-constrained quadratic program if a specific constraint was not included. I would like to know whether ...
0
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1answer
21 views

How to find uppper bound for the hyper parameters in linear model?

I have loss function of the following form: $$argmin_w= ||Y-w^TX||_2^2 + \lambda||w||_1 + \gamma \sum_{g=1}^G||w_g||_1$$ where $w_g$ is group of coefficents. Given $Y$ and $X$, I would like to find ...
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2answers
65 views

Minimise $||Ax - b||_2$ given $x \geq 0$ and $\textbf{1}^Tx = 1$

$$ \min_x ||Ax - b||_2\; \;\text{given }x \geq 0\;\;\text{and}\;\;\textbf{1}^Tx = 1 $$ I am trying to do the above optimization, I was using common Quadratic programming libraries but their speed is ...
0
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1answer
54 views

Silly Quadratic programming

Suppose that I want to minimize the function $x^2$ subject to the contraint $$ ax\leq b, $$ for some $b> 0$. I solved the problem if the contraint is an equality but I'm not sure how to go about ...
0
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1answer
33 views

binary quadratic mixed integer nonlinear programming to inequalities

I have this unconstrained z=x.y where x,y are 0-1 integers. How to reformulate that into set of mixed integer linear inequalities with exactly the same feasible region?
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0answers
69 views

Non-positive-definite quadratic program minimum simplification

I was finding the minimum of a simple quadratic program with equality constraints: $$ m = \min_x x^TQx\qquad\mathrm{s.t.}\quad Ax=b. $$ Using a formulation with Lagrange multipliers from Wikipedia I ...
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0answers
23 views

Constraint breaking in constrained least squares

In a constrained least squares problem $\boldsymbol{y} = \boldsymbol{X}\boldsymbol{\beta},\ \boldsymbol{\beta} = \{\beta_1, \ldots, \beta_n\}$ solved with quadratic programming, suppose that I have ...
0
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1answer
120 views

How to map quadratic programming formulation to dual soft margin SVM

I am trying to use quadratic programming for SVM and I am confused about how to map SVM formulation to quadratic programming formulation given in CVXOPT (Python package). This is what CVXOPT gives us ...
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0answers
25 views

different block size in Block Matrix

I am following a book and having trouble understanding. so from Section 6.6.2 page 496, i have A = NxN matrix b = Nx1 vector c = Nx1 vector from Section 14.3.2 ...
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1answer
29 views

How to Reformulate an Image Constraint in the Dual of a Quadratic Program with Nonconvex Constraints?

I have the following nonconvex optimization problem, for which I want to formulate the dual: $\mathcal{P}:\underset{x}{\text{min}} \quad x^\top A x + b^\top x \\ \quad \ \ \text{s.t.} \quad x\in\{0,1\...
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0answers
29 views

On approximation of maximization of quadratic function over a convex set

Given a convex function $f : \mathbb{R}^n \to [0,\infty)$. Let $L = \left\lbrace x \in \mathbb{R}^n \mid f(x) \leq 1 \right\rbrace$ be it's sub-level set and suppose that $L$ is not empty as well as ...
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0answers
40 views

If the relative value of the most negative eigenvalue is small, can we view the quadratic program as convex?

Consider symmetric matrix $A \in \mathbb{R}^{n \times n}$ with $n-1$ negative eigenvalues and a positive one such that $\lambda_1 > \lambda_2> \dots > \lambda_n$, where $\lambda_n$ is the ...
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0answers
27 views

Quadratic Matrix programming

I would like to find a numerical solution to the following quadratic matrix programming problem. Square matrix A has thousands of rows, so I need a algorithm which works fast. I looked into Python but ...
1
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1answer
121 views

What is the fastest algorithm to solve an equality-constrained convex quadratic program?

I am trying to solve the following convex problem $$\begin{array}{ll} \text{minimize} & x^T Q x + px\\ \text{subject to} & Ax + b = 0\end{array}$$ which arises in the active set method for ...
2
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1answer
103 views

Is the optimal solution of a convex problem continuous with respect to parameters?

Given the convex problem $$\begin{array}{ll} \text{minimize} & \displaystyle\sum_{i=1}^{N} f(x_i)\\ \text{subject to} & Ax = B\\ & 0\leq x_{\min} \leq x_i \leq x_{\max}\end{array}$$ ...
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0answers
44 views

Support Vector Machines: question about the underlying math

I'm new to Support Vector Machines and I've been trying to get into the underlying math (instead of just using Scikit Learn or something like that). I understand the math behind it up to the point ...
1
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1answer
81 views

Solving inequality-constrained least-norm problem non-iteratively

I am looking into the following constrained quadratic program in $x \in \mathcal{R}^4$ $$\begin{array}{ll} \text{minimize} & \| x \|_2^2\\ \text{subject to} & A x = b\\ & x_{\min} \le x_i ...
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0answers
65 views

Checking inequality

Let $\pi$ be a given permutation of the integers $\{1,\ldots,n\}$ and let $$\mathcal{X}=\{x\in\mathbb{R}_{+}^{n} \mid x_{\pi(1)}\geq\cdots\geq x_{\pi(n)},\mathbb{1}^\top x\geq \alpha\},$$ for some $\...
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0answers
117 views

Solving underdetermined linear system using least squares

As in case of linear overdetermined system of equations, we can prove that the cost function i.e. the least square function is convex. But in linear underdetermined system, we know that there exist ...
0
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1answer
111 views

How to increase the accuracy of optimal point obtained by ADMM?

What is the best algorithm to polish the optimal point obtained by the ADMM method for a constrained quadratic program? I've already studied the primal-dual method. Although it can obtain a solution ...
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0answers
38 views

Solving/minimizing $Fw=b$ with $F$ and $w$ unknown and $b$ known

I have a system $Fw=b$ that can be underdetermined, square or overdetermined. The matrix $F$ contains known scalar functions $f_i(t)$ evaluated at unknown values $t_j$. The vector $w$ contains unknown ...
0
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1answer
175 views

Solve Matrix Least Squares (Frobenius Norm) Problem with Lower Triangular Matrix Constraint

Let $\mathbf{A} \in \mathbb{R}^{N \times N}$, $\mathbf{X} \in \mathbb{R}^{N \times M}$, and $\mathbf{B} \in \mathbb{R}^{M \times N}$. We intend to solve for $\mathbf{X}$ by solving the following ...
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0answers
63 views

Primal Dual Interior Point convergence.

I have developed a Primal Dual Interior point algorithm to solve linear inequality constrained Quadratic problems. But sometimes it cannot reduce the residual so as to satisfy the stop criteria. I ...
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0answers
69 views

Equivalent forms Optimization

We were told to assume in class that the below optimization formulations are equivalent- $$\min_w\max_{\delta:||\delta||_F\leq\epsilon}||(X+\delta)w-y||_2^2$$ $$\min_{w}||Xw-y||_2^2+\lambda||w||_2^2 ...
4
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1answer
56 views

Convex quadratic program with linear equality constraint and bounds on variables

$$\begin{array}{ll} \text{minimize} & \sum \limits_{i=1}^n x_{i}^{2}\\ \text{subject to} & \sum \limits_{i=1}^n x_{i} = m\\ & 0 \leq x_i \leq a_i\end{array}$$ I know that if the $a_i$ are ...
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0answers
15 views

Linear optimization under a binary linear classifier constraint with unknown weights

Let $c(x), x\in\mathbb{R}^N$ be a binary linear classifier, i.e. $c(x) = w^\top x + b > 0$, and suppose we have no access to $w\in\mathbb{R}^N$ and $b$ (we can only query the output of the ...
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1answer
48 views

Computing the Lagrange multiplier in constrained optimization problem

As I know the value of the Lagrange multiplier does not have a valuable meaning in most cases. Our goal is to compute the values of the parameters. Here is an example for constrained optimization: ...
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0answers
22 views

How can we transform the objective function of LQG problem as function of the estimation error?

Consider an infinite-horizon classical Linear-Quadratic-Gaussian (LQG) problem \begin{eqnarray*} \underset{x,u}{\min} & & \underset{\eta\rightarrow\infty}{\lim}\mathbb{\mathbb{E}}\left[\frac{...
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0answers
25 views

Minimizing quadratic over region

Let $x_1, ..., x_n\ge 0$ with $x_1+...+x_n=1$. Minimize $$\sum_{j=1}^na_j\left(\sum_{i=1}^nw_{i,j}x_i-y_j\right)^2$$ where $a_j$ and $w_{i,j}$ are some weights. Use $x_n=1-x_1-...-x_{n-1}$. Without ...
0
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1answer
74 views

Strong duality strictly convex quadratic problem

Assume we have this strictly convex quadratic programming: $$f(x) = x^\top A x + b^\top x,$$ $$Ax \leq b$$ $$ 0 \leq x \leq 1$$ Where $A$ is symmetric and positive definite, and the feasible set is ...
0
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0answers
63 views

Dual of a convex SOCP

I'm reading up on convex optimization this summer and wanted to check my understanding of an example problem. $$\begin{array}{ll} \text{minimize} & \| x \|_2\\ \text{subject to} & \...
0
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0answers
32 views

Concave quadratic function unbounded below

$$f(x) = y^\top x - \frac{1}{2} x^\top Q x$$ where $Q$ is both symmetric and positive definite is shown here to be bounded above. However, is the following proof correct for $f(x)$ being unbounded ...