Questions tagged [convex-optimization]

Convex Optimization is a special case of mathematical optimization where the feasible region is convex and the objective is to either minimize a convex function or maximize a concave function.

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4
votes
2answers
8k views

Tikhonov regularization in the non-negative least square - NNLS (python:scipy)

I am working on a project that I need to add a regularization into the NNLS algorithm. Is there a way to add the Tikhonov regularization into the NNLS implementation of scipy [1]? [2] talks about it, ...
3
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2answers
736 views

NP-Completeness

Consider an instance of non-convexoptimization problem: It seems that this problem is NP-complete. How can I find a suitable reduction for this?
3
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1answer
259 views

non-smooth convex c++ solver

I happened to know that there are advanced established techniques for non-smooth convex optimization in research. For example, these two papers: Nesterov, "Smooth minimization of non-smooth functions"...
6
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3answers
2k views

What is the most appropriate derivative free optimization algorithm

We can use random optimization/ derivative free/ direct search to find the minimum of some black box function $f$. If I have some 2D black box function, $f(x,y)$ - which I know to be convex - what ...
2
votes
1answer
938 views

Doubt regarding stopping criterion for Newton method

I am solving an unconstrained convex optimization problem, which can easily have a million variables. I am trying to get a working system with a toy problem of around 200 variables. I am noticing that ...
8
votes
5answers
4k views

Minimizing $\mathrm{trace}(S)+\mathrm{trace}(S^{-2})$ using CVX

In Matlab, I would like to minimize the function $$f(S)=\mathrm{trace}(S)+\mathrm{trace}(S^{-2})$$ where $S \in \mathcal{M}_{m,m}$ is symmetric and positive definite, which is definitely a convex ...
7
votes
3answers
1k views

Nearest positive semidefinite matrix to a symmetric matrix in the spectral norm

So I have a symmetric matrix $A$ and I would like to solve the optimization problem, $$\hspace{2.5mm}\text{Minimize}\;\; \|A-S\|_2$$ $$\hspace{-5mm}\text{Subject to}\;\; S\geq0.$$ $A$ is given and $S$ ...
4
votes
1answer
389 views

Best platform for complex SDPs with n and m around 5-15K?

I am looking to solve a class of SDPs with complex entries, with the semi-definite cone $S^n$, $n$ around 5000 to 15000. Also, $m$, the number of equality/inequality constraints is close to $n$. I ...
3
votes
3answers
940 views

Mathematical programming formulation of triangle intersection

Given variables $a_1$, $b_1$, $c_1$ and $a_2$, $b_2$, $c_2$ representing the vertices of two plane triangles, how might one specify the requirement for the two triangles to intersect as an objective ...
3
votes
2answers
2k views

Subgradients of non-convex functions

In these notes (section 2.3), it is stated that: A point $x^*$ is a minimizer of a function $f$ (not necessarily convex) if and only if $f$ is subdifferentiable at $x^*$ and $0 \in\partial f(x^*).$ ...
3
votes
2answers
182 views

polynomial time solvability

Consider the following optimization problem: $Min \qquad C^TX$ S.t.: $\qquad AX=0;$ $x_ix_j=x_kx_t$$\quad $for some $i\neq j\neq k\neq t$ $X=(x_1,x_2,...x_n)$ and $\quad x_j\geq 0\;\; j=1,2,......
2
votes
1answer
528 views

Reformulate a strictly convex QP problem containing absolute value term

Can the following strictly convex optimization problem be reformulated into a standard form that is also a strictly convex problem? $$\begin{align} &\text{Minimize }\frac{1}{2} x^T Q x + a^T x + ...
1
vote
1answer
1k views

Convex Polygon Intersection

Determining the intersection of two convex polygons is one of the fundamental problems in computational geometry . I'm asking for an algorithm having: INPUT: Given two convex polygons P and Q in 2D (...
1
vote
1answer
440 views

Feasibility checking

Consider the following optimization problem: $Min\;\;\; CX$ $AX\geq b$ $x_ix_j= x_s x_t\;\;\; i\neq j \neq s\neq t$ $x_j\geq 0;$ Where $A$ is the adjacency matrix and $C$ is a constant vector. ...
1
vote
2answers
472 views

What method do you suggest to solve this minimax, quadratic in both variables problem?

I have a problem of the form, \begin{align} minimize_{y} maximize_{x}&\quad x^T y - y^T (B\odot x x^T) y\\ s.t. &x\in [l,u]\\ &Ay=b \end{align} How to efficiently solve this problem? ...
0
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0answers
41 views

state of art non smooth convex optimization [duplicate]

Basically, I am trying to implement non-smooth convex optimization in c++. I am wondering what is the state of art condition of non-smooth convex optimization. For example, what's the best method ...
0
votes
1answer
1k views

How to define the derivative for Scipy.Optimize.Minimize

I am trying to use scipy.optimize.minimize to minimise a quadratic objective function: $f(x) =x^\top Q x$. As a start, I have successfully implemented this using the built-in Nelder-Mead Simplex ...
0
votes
2answers
209 views

Convex quadratic problem solver gives different answers?

I am pretty sure that the following variance objective function should be a convex quadratic problem. My objective function is as follows: $$ \text{argmin } \text{var }(X*L) \xi \geq 1, \text{ where }...
0
votes
0answers
33 views

Minimizing the ratio of two specific non negative quadratic convex functions

$F$ is $m\times m$ diagonal, with real non negative elements $D$ is $n \times m$ complex $P$ is $n \times 1$ complex $A$ is $m \times 1$ complex. Minimize $\Gamma(A)$, with respect to $A$. $$\...