A non-convex optimization problem is one where either the objective function is non-convex in a minimization problem (or non-concave in a maximization problem) or where the feasible region is not convex.

# Questions tagged [non-convex-optimization]

569 questions

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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\{ \mathbf{b}^H…

dineshdileep

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votes

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### Convergence to a local minimum of a nonconvex function

Let $f: \mathbb{R}^n \to \mathbb{R}$ be a differentiable nonconvex function that is bounded below. Let $x^* \in \mathbb{R}^n$ be a local minimum of $f$. Suppose we run gradient descent algorithm $$x^{k+1}=x^k -\alpha \nabla f(x^k)$$ from $x^0 \in…

Sepide

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### What is the range of $\vec{z}^{ \mathrm{ T } }A\vec{z} $?

Let A be a 3 by 3 matrix
$$\begin{pmatrix}
1 & -2 & -1\\
-2 & 1 & 1 \\
-1 & 1 & 4
\end{pmatrix}$$
Then we have a real-number vector $\vec{ z }= \left(
\begin{array}{c}
z_1 \\
z_2 \\
z_3
\end{array}
\right)$ such that
$$\vec{z}^{…

ohisamadaigaku

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votes

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### 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

- 51
- 2

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votes

**2**answers

### Reference on Lipschitz property of the infimum of a family of Lipschitz functions

I can prove the following fact: the infimum, or supremum, of any family of L-Lipschitz functions is L-Lipschitz, as long as the constant L is fixed.
However, since this is a very basic result, I am interested in a reference where it is proved.
Any…

John D

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votes

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### Convert a non-convex QCQP into a convex counterpart

Problem
We consider a possibly non-convex QCQP, with nonnegative variable $x\in \mathbb{R}^n$,
\begin{equation*}
\begin{aligned}
& \underset{x}{\text{minimize}}
& & f_0(x) \\
& \text{subject to}
& & f_i(x) \leq 0, \; i = 1, \ldots, m\\
& & & x \geq…

Mr.Robot

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votes

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### 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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### Minimize $x^T A y$, subject to $ x^Ty\geq 0$, where $A=\Phi^T\Phi$ is symmtric and semi-positive definite.

I try to solve it by KKT conditions.
The Lagrangian is
$L=x^TAy-\lambda x^Ty$.
Its KKT conditions are given by
$$
\begin{align}
Ay-\lambda y=&0\quad (1)\\
A^Tx-\lambda x=&0\quad (2)\\
\lambda\geq &0\quad (3)\\
x^Ty\geq &0 \quad (4)\\
\lambda…

Hao WANG

- 75
- 4

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### Nearest Semi-Orthonormal Matrix Using the Entry-Wise $ {\ell}_{1} $ Norm

Given an $m \times n$ matrix $M$ ($m \geq n$), the nearest semi-orthonormal matrix problem in $m \times n$ matrix $R$ is
$$\begin{array}{ll} \text{minimize} & \| M - R \|_F\\ \text{subject to} & R^T R = I_n\end{array}$$
A solution can be found by…

Francis

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**0**answers

### Overdetermined linear system with norm equality constraint

Consider the simple optimization problem
$$\min_x \|b-Ax\|_2 \ \ \ \ \ \ \text{subject to} \ \ \ \ \ \ \|x\| = 1$$
where $b\in\mathbb{R}^{N}$, $A\in\mathbb{R}^{N\times 3}$, $x\in\mathbb{R}^{3}$ with $N > 3$ (overdetermined system). I need to solve…

GDumphart

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### Global optimum of sum of convex functions

Take two real differentiable convex functions, $f_1$ and $f_2$, defined on the unit interval $[0, 1]$. I want to find the global optimum of
$$\min_{x \in [0,1]} a f_1(x) + b f_2(x)$$
for given $a, b \in \mathbb{R}$. Is there a simple solution to…

user38397

- 313
- 2
- 13

**5**

votes

**1**answer

### Maximizing a quadratic function subject to $\| x \|_2 \le 1$

Consider the $n$-dimensional quadratically constrained quadratic optimization problem
$$\begin{array}{ll} \text{maximize} & \frac12 x^T A x + b^T x\\ \text{subject to} & \| x \|_2 \le 1\end{array}$$
where $A$ is a symmetric $n\times n$ matrix that…

user856

**5**

votes

**1**answer

### What is the definition of "convex relaxation" in clustering?

I have following text from a paper i am trying to understand:
I don't understand what does below sentence refers to as being convex/non-convex
The problem is that even though the objectives (1) and (2) are convex the constraint that K is valid…

dariush

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votes

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### Non-trivial example of a local optimum without satisfying Karush-Kuhn-Tucker (KKT) optimality conditions

I am looking for an example of a locally optimal point of the nonlinear program:
$$\{ \min f(x) , g_i(x) \geq 0 \}$$
that does not have a singleton feasibility set.
e.g. not the example of $\{ \min x, x^2\leq 0 \} $ found here where the feasibility…

shnnnms

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### Optimal value in $\max_{x} \max_{y} f(x,y) = \max_{x,\ y} f(x,y)$

I am trying to understand a few things about sequential and simultaneous optimization in [1]. In this post, it is shown that
$$\max_{x} \max_{y} f(x,y) = \max_{x,\ y} f(x,y).\tag{1}$$
Thanks to @Shiv Tavker comment, I understand that in order to get…

darkmoor

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