0
votes
1answer
47 views

Solving LP with two $L_1$ inequality constraints

Is there a "fast" way to solve the following LP formulation with the following constraints: $$ \max_{\mathbf{f}} \mathbf{f}'.\mathbf{g} \\ \mathbf{1}'\mathbf{f}=1\\ \|\mathbf{f}-\mathbf{h}\|_1\le ...
0
votes
1answer
27 views

closed form vs gradient descent baseed methods

I am a beginner to optimization. Could anybody give me a simple example to illustrate when I should use closed form and when I should use iterative methods like gradient descent? Thanks in advance.
0
votes
1answer
28 views

how to check an optimization function is convex or not

This is the sparse coding optimization function: $\operatorname*{argmin}_{B, \alpha} \sum_j \| \bf{x}_j - B\bf{\alpha}_j \|_2^2 + \lambda\sum_j |\bf{\alpha}_j|_1$ I read in the literature that this ...
0
votes
1answer
21 views

how to differentiate to optimize this function?

I have an optimization function in the following form: $E = \operatorname*{argmin}_{A} \sum_j \| A\bf{x}_j - B \|_2^2 + \mu\sum_i a_{ii}^2$ Where, A is an unknown diagonal matrix with elements ...
1
vote
2answers
51 views

How to solve this optimization problem? (may be gradient descent?)

I have the following optimization problem. $$\operatorname*{argmax}_{w} \|(1-w)\boldsymbol{X} -w\boldsymbol{Y}\|^2 \\ s.t. \quad 0<w<1 $$ How can I find the solution of this problem? May be ...
1
vote
2answers
34 views

multi-objective optimization

I am currently encounterring a optimization problem. The goal is optimize an objective function A and B at the same time. But the problem is that optmizing A will almost always tradoff with B, such ...
1
vote
1answer
52 views

Matrix Maximization

I would like to solve the following optimization problem for a matrix $X$ which is symmetric and positive-semidefinite: $$ \mathrm{maximize} \, \, \, f(X) = \log \mathrm{det} X - k_1 \log(k_2 + a^T X ...
0
votes
2answers
56 views

Regularization vs. Inequality Constraint

For what values of a regularization parameter $\alpha$, there is an equivalent inequality constraint in convex optimization? In particular, in the convex optimization problems below $$ \text{ Problem ...
1
vote
1answer
34 views

Linear constraint in convex optimization

Is it true that the solution to a linearly constrained convex minimization problem can only be placed on the boundary of the constraint set, for any nonlinear convex objective, e.g. $$ \min_x f(x)$$ ...
1
vote
1answer
57 views

KKT point of a constrained optimization problem

Min$_{x}~x$ Subject to $x \geq 0$ For this problem, is $(x^{*}, \lambda^{*})=$$(0,0)$ a KKT point ? My try : I formulated corresponding Lagrangian and tried to find out the KKT point(s). ...
0
votes
0answers
35 views

Linear least squares with sparse inequality constraints for support function estimation

The initial problem is the following: $$ ||h - h^{0}|| \to min \; \; s.t. Qh \leq 0 $$ where $h^{0} \in \mathbb{R}^{n}$ is known vector and $Q$ is a $m \times n$ matrix. The problem arises in specific ...
0
votes
1answer
64 views

Cyclic monotonicity of sub-differential domain and convex property

I am looking for hints/proof's overview/reference about this proposition : Let $S\subset \mathbb{R}^d\times\mathbb{R}^d$. There exist a convex function $\phi$ such that $S\subset \partial\phi$ ...
1
vote
1answer
126 views

Free software or algorithm for Second-Order Cone Program

I need to solve the following optimization problem: $$ \mathbf{x}^\ast = \operatorname{argmin}_{\mathbf{x}} \Vert \mathbf{Rx} \Vert_2^2 \;\;\; \mathrm{s.t.} \;\;\; \mathbf{s}^\mathrm{H} \mathbf{x} = ...
0
votes
1answer
49 views

How many methods could be used to solve this optimization problem with equality constraints?

I wonder whether there is a simplest method for this problem. The function to maximize is $F(x)$. $F(x)=\|Kx\|_2^2=x^TK^TKx$, where $K\in \mathbb{R}^{n\times d}$ and $x\in \mathbb{R}^d$. and $\nabla ...
1
vote
1answer
74 views

Linear System with constrained solutions

After a model my problem I found a rectangular linear system : $$Ax=b$$ I can easely solve it with a least square with QR/SVD... But the model include constrains for each solution $x_i$, the $\vec{x}$ ...
1
vote
2answers
67 views

When $\min_{x \in X,y \in Y} f(x,y) = \min_{x \in X} \min_{y \in Y} f(x,y)$?

When $$ \min_{x \in X,y \in Y} f(x,y) = \min_{x \in X} \min_{y \in Y} f(x,y) \qquad? $$ I mean when we are minimizing a function with respect to two variables, under what conditions we are allowed to ...
3
votes
1answer
60 views

Rewrite constrained optimization objective

I wanted to ask, under which conditions can one rewrite the optimization objective $\min_x f(x)\;\;\;s.t.\;\;\;g(x) \leq s$ as $\min_x g(x)\;\;\;s.t.\;\;\;f(x) \leq t$ I have particular interest ...
1
vote
1answer
133 views

Sparse least-squares fitting of discrete probability distributions

I have an optimization problem involving $n$ discrete probability distributions and I am looking for a suitable solution for this problem that I can implement as computer program. Let the vector ...
3
votes
0answers
102 views

Does convexity of a function guarantee tractability of finding its minimum?

Formulating a problem as a convex optimization problem usually implicitly considered to imply being able to find global minimizer of the objective. My question is that if it is true or not. ...
0
votes
2answers
405 views

What Stopping Criteria to Use in Projected Gradient Descent

Suppose we want to solve a convex constrained minimization problem. We want to use projected gradient descent. If there was no constraint the stopping condition for a gradient descent algorithm would ...
0
votes
1answer
29 views

unstable optimizer, stable objective

I am trying to minimize a convex objective numerically using gradient descent. I select the starting point randomly. I repeat the experiment multiple times. The optimal objective value I get each time ...
1
vote
1answer
1k views

what are Smooth and Non Smooth Problem in optimization?

I am trying to understand the difference between the optimization problem types which are basically smooth and non smooth. I also found this question what does a smooth curve mean? I understand that ...
0
votes
2answers
259 views

Portfolio Optimization Problem Without Correlation Info

I received this interesting problem from a friend today: Assume that you are a portfolio manager with $10 million to allocate to hedge funds. The due diligence team has identified the following ...