# Questions tagged [convex-optimization]

Convex minimization, a subfield of optimization, studies the problem of minimizing convex functions over convex sets. The convexity property can make optimization in some sense "easier" than the general case - for example, any local minimum must be a global minimum.

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### Subtracting Values from a Positive semidefinite Matrix in a Semidefinite Program

I'm trying to construct an SDP relaxation for a non-convex quadratic program ($x^{\intercal}\mathbf{H}x$) with the following objective function: x_{11}y_{11} + x_{12}y_{12} + x_{21}y_{...
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### How to simplify the following constraints as I'm using MIP optimization solver in python?

Following is the initial snippet of the code: ...
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### Non-symmetric Positive Definite/Semidefinite Matrix in Quadratic Program

A necessary condition in any quadratic programming to be convex is the matrix $\mathbf{Q}$ in the formulation $x^\intercal \mathbf{Q}x$ to be positive definite or positive semidefinite. Positive ...
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### Black-box optimization of a single parameter function with high cost evaluation

I need to solve a series of single parameter black-box minimization problem. The underlying cost functions are quite simple. They always have the same shape: a global minimum inside a fixed interval (-...
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I'm totally new to the world of optimization and I have an optimization problem that I think it can be formulated as Mixed Integer Quadratically Constrained Quadratic Program (QCQP) but I'm not sure ...
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### Existence of a transformation to convex optimization

Question Does a transformation of the following problem to convex optimization exist? \begin{aligned} \label{1} \min_{\vec{x}, \vec{y}} \quad & F(\vec{x}, \vec{y}) \\ \textrm{s.t.} \quad ...
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### How to make following constraint a convex one?

I would like to write a constraint as follows, where $x,y>0$ are optimization variables, and $a,b,c,d,A$ are positive constants. How to make it a convex constraint? \frac{{ax}}{{\...
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### What is a good way to penalise LP relaxation?

I have a binary integer program. It is of a large size and the solver is taking longer time. I am thinking of relaxing the binary integer variable and making it a continuous variable. How can I ...
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### Model definition in pyomo to solve online optimization problem

I am trying to model the attached online optmization problem in pyomo. Eventually, I am going to use the octeract solver to find the matrix soluions of W and G. I would like to ask advice about ...
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### Prove that $x^*$ is an optimal solution where $f_0$ is $C^1$ and convex and $f_i$ are $C^1$ and strictly convex functions

Let $x^*$ be a feasible solution of the following convex optimization problem \begin{align}\min&\quad f_0(x)\\\text{s.t.}&\quad f_i(x)\leq0,i=1,\ldots,m\end{align} where $f_0$ is $C^1$ and ...
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### Prove Non-Homogeneous Farkas' Lemma

Let $A\in\mathbb{R}^{m \times n}, c\in\mathbb{R}^{n}, b\in\mathbb{R}^{m}, d\in\mathbb{R}$. Suppose that there exists $y\geq0$ such that $A^Ty=c$. Question: prove that exactly one of the following is ...
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### active set method guaranteed convergence

I'm using Active Set Method to solve a nonlinear optimization function minimizing a convex function over a polyhedron of 2 linear inequalities starting at an interior point $x_o$ At this point is it ...
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### How to model these constraints correctly

$W$ is a vector of $N$ complex elements. $D$ is a binary variable The requirements are: when $D==1$, $L_{\min}\le ||W||_2^2\le L_{\max}$ and when $D==0$, $||W||_2^2=0$ I have formulated the following ...
Let $\mathcal{U} = \{ [x_1, ..., x_n] \in \mathbb{R}^n : 0 \leq x_i \leq 1\}$ be the unit hypercube and $C \in \mathbb{R}^n\setminus\mathcal{U}$ fixed. Let us consider the following problem  \max_{X ...
I wondered about this question for sometime. Definition of Stationarity (P) $\mbox{min} f(x)$ s.t $x\in C$ Let $f$ be $C^1$ function over a closed and convex set $C$ . then $x^*$ is called a ...