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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### positive constant divided by a concave function, how to convexify this constraint?

I have following constraint in my optimization problem:
$0 \leq t \leq \frac{S}{B \ R(p)}$
where $t,p$ are optimization variables and $S,B$ are positive constants.
$R(p)$ is a concave function which is in the denominator, is there any way to make…

Syed

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### Non Convex Optimization update analytic proof

I need help with the below proof,
$$A = \underset{A}{\text{argmin}}(\frac{1}{2} ||X_{(1)} - A(C \odot B)^T||^2_F + ||\Lambda \boxdot (A - \tilde{A})||_F^2 + \frac{\rho}{2} ||A - \tilde{A}||_F^2) \\ = (X_{(1)}(C \odot B) + \rho \tilde{A} - \Lambda )…

Mour_Ka

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### Maximization of quadratic function

Consider the following maximization problem.
$$
\begin{aligned}
&\max_{x \in \mathbb{R}^n}x^TPx\\
&a_i \leq x_i \leq b_i,~ \forall i \in \mathbb{Z}_{[1,n]}\,,
\end{aligned}
$$
where $x_i$ is the $i$-th scalar component of vector $x$ and $P>0$ is…

pulosky

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### Convex function - Involves powered quadratic form

I would like to prove/disprove that the following function is convex with respect to $w$:
$$
\sum_{i=1}^{N}\ln[1+(w^{t}X_iw)^{-y}]
$$
where I have N data points, $w$ is the $p\times 1$ vector, $X_{i}$ is a $p \times p$ diagonal matrix and $y$ is…

Tuan Do

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### Find the very first feasible solution (critical point nearest to initial point in constrained optimization)

While in optimization problems, usually, the goal is to find an extremum, I'm interested in finding the first feasible solution that satisfies all constraints (i.e. the nearest critical point to my initial point). In other words, I want to move from…

Chris Sherwood

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### If these problems are equivalent how can I prove that mathematically?

If $\alpha\,\, and\,\, p$ are non-negative scalars, are the two problems given below equivalent? I know that the values of objective functions will be different but I want to know will they result in same values of $\alpha\,\, and\,\, p$.
If they…

Zain Ali

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### Minimize $\sum_{i=1}^M x_i e^{y_i}$

everyone. I want to minimize an objective function
\begin{equation}
\min_{\mathbf{x},\mathbf{y}} \ \sum_{i=1}^M x_i e^{y_i},
\end{equation}
where $\mathbf{x},\mathbf{y}\in \mathbb{R}^M$. As far as I know, the objective function is non-convex,…

user350491

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### Proving tightness of a semidefinite relaxation of a nonconvex quadratic program

The Lagrangian of my semidefinite relaxation of a nonconvex QCQP (quadratically-constrained quadratic program)…

HD Lang

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### How to solve this quadratically constrained quadratic programming problem?

Could you please shed some lights on this? (Not a homework problem)
I am looking for solutions of the following problem in $b$:
$$\max \| X b \|_2 \quad\text{subject to} \quad \| b - b_0 \|_2 < a, \| b \|_2 = 1$$
where matrix $X$, $a$ and $b_0$ are…

Luna

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### solving set a set of in-equations for x and y

I have a set of below in-equations. Can we solve for a solution of x and y
$$x>2$$
$$xy>5$$
$$y^2x\leq9$$

GopalaRaok

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### How can I mathematically prove this?

How can I mathematically prove that P1 and P2 will have the same value of $\eta$ at optimality? Although it seems clear from the intuition. I am looking for proof in the language of mathematics, not in English.
P1
\begin{align}
\max_{\eta,x_1,x_2}…

Zain Ali

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### How does a non-convex optimization problem can be solved to get a near optimal solution?

I have an optimization problem of the form $f(x_1,x_2,y_1,y_2) = \frac{y_1}{x_1} + \frac{y_2}{x_2}$ for all real values of $x_1,x_2,y_1,y_2$ . Also $f$ is found to be non-convex in nature. I want to minimize this non-convex function $f$. How can I…

thesukantadey

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### How to determine minimum point or maximum point

Given below
$2y^2 + x^2$
Such that $x + y =1$
How do I show that it is a minimum or maximum point.
What I have tried:
I differentiated wrt x

Favour Chukwuedo

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### Proving quasiconvexity of $x^3 + y^3$

I was studying quasiconvex functions, and suddenly I have stumbled upon the function $f(x,y)= x^3+y^3$.
Is it quasiconvex? From the graph it seems like.

Dipan Banik

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