Mathematics

Questions tagged [qcqp]

A quadratically constrained quadratic program (QCQP) is an optimization problem in which both the objective function and the constraints are quadratic.

205 questions
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votes
7 answers

Maximize the value of $v^{T}Av$

Let $A$ be a symmetric, real matrix. The goal is to find a unit vector $v$ such that the value $v^{T}Av$ is maximized, and minimized. The answer is that $v$ should be the eigenvector of $A$ with largest eigenvalue, and smallest eigenvalue. Could…
user4205580
  • 2,233
14
votes
3 answers

Solve least-squares minimization from overdetermined system with orthonormal constraint

I would like to find the rectangular matrix $X \in \mathbb{R}^{n \times k}$ that solves the following minimization problem: $$ \mathop{\text{minimize }}_{X \in \mathbb{R}^{n \times k}} \left\| A X - B \right\|_F^2 \quad \text{ subject to } X^T X =…
Alec Jacobson
  • 573
13
votes
8 answers

Distance of ellipse to the origin

Calculate the minimum distance from the origin to the curve $$3x^2+4xy+3y^2=20$$ The only method I know of is Lagrange multipliers. Is there any other method for questions of such type? Any help appreciated.
Shobhit
  • 6,970
12
votes
3 answers

Trace minimization with constraints

For positive semi-definite matrices $A,B$, how can I find an $X$ that minimizes $\text{Trace}(AX^TBX$) under 'either' one of these constraints: a) Sum of squares of Euclidean-distances between pairs of rows in $X$ is a constant $\nu$. or b) $X$ is…
qlinck
  • 1,191
11
votes
4 answers

Minimizing a quadratic function subject to quadratic constraints

Okay, so I am attempting to minimize the function $$f(x,y, z) = x^2 + y^2 + z^2$$ subject to the constraint of $$4x^2 + 2y^2 +z^2 = 4$$ I attempted to solve using Lagrange multiplier method, but was unable to find a $\lambda$ that made the system…
user345
  • 791
10
votes
2 answers

Eigenvalue bound for quadratic maximization with linear constraint

This builds on my earlier questions here and here . Let $B$ be a symmetric positive definite matrix in $\mathbb{R}^{k\times k}$ and consider the problem $$\begin{array}{ll} \text{maximize} & x^\top B x\\ \text{subject to} & \|x\|=1 \\ & b^\top x =…
sven svenson
  • 1,440
8
votes
1 answer

Minimizing Quadratic Form with Norm and Positive Orthant Constraints

Let $ M $ be a positive semi definite matrix. I want to solve $$ \arg \min_{x} {x}^{T} M x \quad \mathrm{s.t.} \quad \left\| x \right\| = 1, \ x \succeq 0 $$ where $ x \succeq 0 $ means each coordinate of $x$ is nonnegative. Is there a standard…
user7530
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7
votes
1 answer

How can I experiment with Lagrange multiplier in QCQP?

Suppose we want to solve following optimization problem (it is a PCA problem in this post ) $$ \underset{\mathbf w}{\text{maximize}}~~ \mathbf w^\top \mathbf{Cw} \\ \text{s.t.}~~~~~~ \mathbf w^\top \mathbf w=1 $$ As mentioned the the post, using the…
HXD
  • 493
6
votes
2 answers

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 worked for me thus far: $$ \begin{equation}…
Kirillvh
  • 193
6
votes
1 answer

SDP relaxation of non-convex QCQP and duality gap

Short version Is there a duality gap between a QCQP problem and the SDP problem obtained through Lagrangian relaxation? A paper I'm studying is using this fact, but I cannot achieve the authors' results. Longer version I've been trying to reproduce…
user73053
  • 61
6
votes
2 answers

Least squares problem with constraint on the unit sphere

It is easy to find the minimum of $\|Ax-b\|_2$, when $A$ has full column rank. But how is the case when we add the constraint $\|x\|_2=1$? Or, to be explicit, $$\min_{\|x\|_2=1}\|Ax-b\|_2=?$$ My idea is to construct the corresponding Lagrange…
Gabriel
  • 61
6
votes
4 answers

Quadratic optimisation with quadratic equality constraints

I would like to solve the following optimisation problem: $$\text{minimize} \quad x'Ax \qquad \qquad \text{subject to} \quad x'Bx = x'Cx = 1$$ Where $A$ is symmetric and $B$ and $C$ are diagonal. Does anyone have a suggestion for an efficient way of…
user111950
  • 565
5
votes
1 answer

Least-squares problem with orthonormality constraints

Given $y_1,\dots, y_n \in {\Bbb R}$ , $w \in {\Bbb R}^d$ , and $x_1,\dots, x_n \in {\Bbb R}^D$ , how do we solve the following optimization problem \begin{align} \min_A \,\, \sum_{i=1}^n \left( y_i - w^T A^T x_i \right)^2\\ \text{subject to} \qquad…
gmravi
  • 386
5
votes
1 answer

Minimizing quadratic function subject to quadratic equality constraint

Given $N \times N$ positive (semi)definite matrix $\mathbf{A}$ , vector $\mathbf{b} \in \Bbb C^N$ and $c > 0$ , $$\begin{array}{ll} \underset{\mathbf{x} \in \mathbb{C}^N}{\text{minimize}} & \mathbf{x}^H\mathbf{A}\mathbf{x} + 2 \Re\left\{…
dineshdileep
  • 9,133
5
votes
2 answers

Minimize $x^T A x$, subject to $\|x\|=1$. Show that ${x^*}^TAx^*$ is the smallest eigenvalue of $A$ in magnitude

I'm solving constrained optimization exercises for preparing my final exam. I got stuck at this question. $$ \begin{array}{ll} \text{min} & \mathbf{x}^T\mathbf{A}\mathbf{x} \\ \text{s.t.} & \|\mathbf{x}\|_2=1 \end{array} $$ I'm asked to show that,…
hkBattousai
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