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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1answer
27 views

Reentrant constraints in active set algorithm?

Problem definition Supposing you're trying to solve a quadratic program: $$ \min_x f(x) = \frac{1}{2}x^T Q x + c^T x \\ \mbox{s.t} \, \; A x \ge b$$ Where Q is square ($n$x$n$), positive semi ...
-2
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0answers
21 views

Quadratic programming with linear equality constraints in Matlab [on hold]

I have to identify an ARX under some linear constraints, this means that I have a quadratic programming with linear equality constraints problem. One way is to use the following equations in the ...
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0answers
10 views

Convex Quadratic Programming - solution always exists?

Assume I want to minimize $$ f(x) = x^T Q x + c^T x $$ $$ Ax \leq 0 , Bx = d$$ where $Q$ is symmetric and positive definite. This implies that f is strictly convex and has at most one minimum. How ...
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1answer
33 views

How to linearize a quadratic objective function with linear constraints?

I have an optimization problem that I'm working on. The objective is defined as follows: $Maximize: c_i\cdot w_i \cdot x_i - d_i \cdot y_i \cdot \delta_i $ subject to some linear constraints where ...
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0answers
16 views

Quadratic programming using Python

guys I'm trying to solve quadratic programming problem with constraints. I know how to solve simple quadratic problems using scipy.optimize like following: Define objective function as F = ...
1
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1answer
28 views

Quadratic Problen with 2 constraints

Could someone help me to solve the following: $\min x^Tx$ s.t. $x^T a=1$ $x^T b=0$ where $x$,$a$ and $b$ are $(N\times1)$ vectors and $1$ and $0$ scalars. Thank you!
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0answers
31 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 ...
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0answers
15 views

First and second derivatives of barrier term in a quadratic programming problem

I am implementing an algorithm of Dang and Xu's, ``Non-convex Quadratic Programming Problem with Box Constraints'' and I'm hoping that somebody could verify what I'm doing. Their algorithm minimizes ...
2
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1answer
39 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
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2answers
106 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 ...
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0answers
16 views

Restating optimization problem for quadratic programming

I'm working on implementing an author disambiguation algorithm as described in Torvik et al's paper. I've got most steps done, but am completely stumped on implementing a quadratic optimization step. ...
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0answers
16 views

Backwards quadratic programming to infer Q matrix

Consider the standard QP problem: $\arg\min \frac{1}{2}x^TQx +c^Tx$ Say I know the optimal $x$ for a large number of solutions to this problem with various (known) $c$, and identical (unknown) $Q$. ...
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0answers
14 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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1answer
51 views

A particular quadratic minimization problem

Given $n^2$ constants $a_{11},a_{12},\ldots,a_{1n},a_{21},\ldots,a_{nn}$ and $n^2$ non-negative variables $x_{11},x_{12},\ldots,x_{1n},x_{21},\ldots,x_{nn}$. Find the minimum value of $$\sum_{i=1}^n ...
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0answers
5 views

Perturbation of Polyhedral Projection

I am interested in understanding the behavior of the Euclidean projection $\pi_K(x)$ as the polyhedral set $K$ varies. I know there are different approaches to this, but for what I am doing it would ...
0
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0answers
32 views

enforcing big M constraints in quadprog

I have a QP with LC type problem. I have several sets of variables x1, x2 ... ,y where x is continuous and y is binary. However I don't enforce this, it is simply bound 0 <= y <= 1 Now I use big ...
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0answers
23 views

Writing down the KKT optimality conditions

Consider the problem Minimize $(1/2)\times{x}^{T}\times Q\times x+{P}^{T}\times x$ Subject to $(1/2)\times {x}^{T}\times P\times x+{d}^{T}\times x≤r$ Where Q and P are n×n matrices, P is ...
0
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1answer
45 views

Homogeneous non-negative least-squares

I would like to least-squares-"solve" a set of linear equations ($\underset{\mathbf{x}}{\mathrm{argmin}}\; \|\mathbf{Ax-b}\|_2$). In my case, $\mathbf{b=0}$, e.g. the system is homogeneous. I also ...
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0answers
30 views

How to find maximum of $w_1^Ta + w_2^Tb + w_3^Tc$

I have a question that is not homework, but I have gotten nowhere on yet. Define: $w_1^T := [x_1,\ldots,x_n]$, $w_2^T := [x_1^2,\ldots,x_n^2]$ and $w_3^T := [x_1^3,\ldots,x_n^3]$. How do I maximise ...
0
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1answer
99 views

How do you minimize “hinge-loss”?

A lot of material on the web regarding Loss functions talk about "minimizing the Hinge Loss". However, nobody actually explains it, or at least gives some example. The best material I found is here ...
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0answers
19 views

Optimization with intervals

I am trying to solve a specific problem, and I was able to summarize it in the following optimization problem. I have a portfolio comprised of two assets. Asset 1 has return $r_1$, standard deviation ...
1
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2answers
84 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 ...
1
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1answer
29 views

Equality constrained Quadratic Program

Consider the QP $$ x^* = \arg \min_{\displaystyle x \in \mathbb{R}^n{\geq 0}} \ \frac{1}{2} x^\top P x + q^\top x \ \text{ sub. to: } A x = b, $$ where $P \succ 0$. Without the non-negativity ...
0
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1answer
58 views

Linear programming with non-convex quadratic constraint

Could anyone let me know if the following linear programming problem can be solved in polynomial time or should be NP-hard? $\min c^Tx$ s.t. $x^TQx\geq C^2, x\in [0,1]^n,c\in ...
0
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1answer
42 views

Is a given point P outside a given bounding box, in Ax < b form

Given a point $x$ and a bounding box $B$ - let's say we have the unit normals $N_i$ of the sides (pointing inwards) and one point on each side $P_i$ - we can check if $x$ is inside $B$ as follows: ...
1
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1answer
45 views

Solving multiple $L_1$ penalties with quadratic programming

Starting from a simple $L_1$ penalization: \begin{equation} min_x \frac{1}{2}||y-x||^2_2 + \lambda||Dx||_1 \end{equation} We can solve this with quadratic programming via the dual problem: ...
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0answers
54 views

a question about relationship between KKT matrix equation and optimal solution of quadratic problem.

I have a question regarding how the KKT matrix plays in solving for optimization problem: Is it correct that the optimal solution for quadratic optimization problem with positive definite hessian ...
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0answers
58 views

Are box constraints problematic when using KKT conditions to solve quadratic programming problems?

I've dealt with quadratic programming before, but I've never seen something of this sort: $$\min \frac12 \|v\|^2 + \sum_ip_i$$ $$\text{s.t. }f(v,p)\ge0 $$ $$0\le p_i\le a$$ for some constant a ...
1
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1answer
167 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 ...
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0answers
17 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} $ ...
0
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0answers
31 views

Implementing SVM: Help converting equation into form of another

I'm currently programming a simple linear SVM (Support Vector Machine). For the optimization involved, I need to find a way to convert the equation $\sum\limits_{i=1}^L a_i ...
1
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2answers
52 views

Trace of quadratic function with 2 PSD matrices - convex?

If A & B are positive semi-definite, is this always convex: $$ trace(XAX^TB) $$ There was a similar question asked here: Trace of a quadratic function, Convexity and here: Confusion related to ...
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0answers
22 views

How to check if steepest gradient method will converge?

So I have this function $ f(x,y) = x^4 - 2x^2 +x + 4y^2 $ and I want to know if the steepest gradient method will converge if I pick an arbitrary point and apply said method. My initial thought ...
0
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1answer
41 views

How can I find the minimum and maximum?

Lets have the following equation: $f(x,y,y) = cos(x)^2+\frac{1}{1+x^3}+y^3+z$ I would like to find the minimum and maximum where $-2<x<2$ $-1<y<1$ $-2<z<1$ How can I do that, I ...
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0answers
36 views

Total unimodularity in quadratic programming.

I have a quadratic integer problem of the following form: \begin{align} minimize & \quad \tfrac{1}{2} x^T Q x + c^T x \\ subject \ to & \quad M x = 1 \\ & \quad x_i \in \{0, 1\} ...
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0answers
39 views

Finding an optimal set of weights for combining correlated classifiers

In order to combine classifiers that are correlated with one another, I would need to solve the following optimization problem: Find a vector $\mathbf{w}$ that minimizes $\mathbf{w}^T M \mathbf{w}$ ...
1
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0answers
49 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 ...
0
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1answer
60 views

Minimization problem convex set

I'm trying to minimize the function: $$f(w)=w^T\mu+k\sqrt{w^T\Sigma w}$$ where $w$ is a vector in $W=\{x \in \mathbb{R}^n|x_1+...+x_n=1 , x_i \geq 0 \forall i\}$. The vector $\mu \in \mathbb{R}^n$, ...
0
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0answers
197 views

Minimum Curvature Path - implementation

I need to implement an algorithm which solves the Minimum Curvature Path problem. I found an interesting solution in this paper http://i.imgur.com/2EeGQKp.png. First, I decided to implement an ...
2
votes
1answer
150 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 ...
2
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0answers
57 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 ...
3
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2answers
81 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 ...
0
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1answer
58 views

How to re-parametrize for quadratic minimization?

Given a real-rectangular matrix $S$ and inorder to solve this simple quadratic programming problem: Minimize $w'S'Sw = \|S w\|^2$ over $w$ subject to $e^Tw = 1$ and $w \geq 0$ using a solver I ...
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0answers
124 views

Optimization problem given a known solution space

Here is my problem. I have to find four points in 3D (x1,y1,z1; x2,y2,z2; x3,y3,z3; x4,y4,z4) that satisfy some given quadratic constraints. In addition, I have a solution space in the form of a set ...
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0answers
75 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 ...
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0answers
77 views

Is this a quadratic programming problem?

I have $N$ unknown 2D points ${\bf x} = (x,y)$ and I want to minimize: $f({\bf x}) = \sum_{i=0}^{N-1}{(||x_{i+1} - x_i||_2 - c_i)^2}$ where c are $N-1$ known scalars. What is the simplest form that ...
2
votes
1answer
169 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} ...
3
votes
1answer
225 views

Optimization problem with ratio objective

I need to solve the following optimization problem $$ \text{maximize} \quad \frac{(a^T x)^2}{x^TBx+c^T|x|} \quad \text{subject to} \quad \|x\|_1=1 \quad (\text{or alternatively} \quad c^T|x|=1), $$ ...
3
votes
2answers
312 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} ...
1
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0answers
86 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}\} \\\ ...