12
votes
Accepted
Solver for convex optimization with exponent in the objective function
You are missing some details for the claimed convexity of $f(x)$. I will assume that you meant to say convex on the domain $C+mx+\frac{s}{x+t} \geq 0$ and $x+t \geq 0$. In this case the epigraph of $f(...
- 1,347
9
votes
Accepted
Determining the optimize lambda in Multi-Objective Optimization
There is no mathematical way to derive (or justify) a value for $\lambda$. The justification has to be made in the context of a specific problem and a specific (reasonably credible) decision maker. ...
- 37.8k
8
votes
Determining the optimize lambda in Multi-Objective Optimization
Another approach could be generating the Pareto Frontier, solving the problem several times for different values of lambda, using a Weighted sum algorithm (see this or this).
8
votes
Accepted
How to find the index of the item, the first time appears?
Here's a formulation if at least one $x_i$ must be $1$:
\begin{align}
\sum_i y_i &= 1 \tag1\label1\\
y_i &\le x_i &&\text{for all $i$} \tag2\label2\\
y_i &\le 1-x_j &&\text{...
- 30.3k
7
votes
Determining the optimize lambda in Multi-Objective Optimization
In addition to the above answers, there's good deal of discussion here. One of the experts logically breaks down some key questions like avoiding dominating solution by sticking to single combination ...
- 3,343
7
votes
How to find the index of the item, the first time appears?
Suppose the index is 1-based and set constant $u_0 = 0$. With binary variables $u_i, y_i, i=1,\dots,n$ and constraints
$$
\begin{align}
u_i &\geq x_i\\
u_i &\geq u_{i-1}\\
u_i &\leq x_i + ...
- 1,036
6
votes
Solver for convex optimization with exponent in the objective function
Use Ipopt and a modeling language. For example, JuMP: https://jump.dev/JuMP.jl/stable/tutorials/nonlinear/introduction/
...
- 921
5
votes
Accepted
Simplest Quadratic Programming algorithm for teaching
You can optimize convex equality-constrained QP by solving a linear equation system (the KKT conditions). This is good knowledge to disseminate and easy for undergrads to grasp in a lecture. Moreover, ...
- 1,347
5
votes
Solver for convex optimization with exponent in the objective function
Best way is to apply log to the function as optimal solution to log of a function is the same for the function.
$ \log f = p[\log (C(x+t)+ mx(x+t)+s)-\log(x+t)]$
Then you have to get the actual ...
- 3,343
5
votes
Is it possible to express these constraints with basic cones?
Let $x_i = \frac{z_i}{y_i}$.
Then, presuming $x_i$ does not appear in any of the other constraints, this appears to be a Generalized linear-fractional programming problem per section 4.3.2 "...
- 13.1k
5
votes
When Biconvex function is Pseudoconvex function?
I am a bit confused by the wording. The title says when biconvex is pseudoconvex, but in the description asks whether a biconvex function is pseudoconvex. I am answering assuming you are asking the ...
- 1,458
5
votes
Accepted
Question About Fritz John Theorem and Slater Constraint Qualification
Lets break down the statement:
There is a ball $U$ about $\mathbf{x}^*$ in $\mathbb{R}^n$.
That is, let $U = \{ x \in \mathbb{R}^n : \| x - \mathbf{x}^* \| \leq \rho \}$ be a norm-ball with some ...
- 1,347
5
votes
Quadratic optimisation with $\ell_1$ constraints with CVXPY
cp.norm(w, 1) == 1 is a nonlinear equality constraint, hence violates DCP rules. It is a non-convex constraint.
Unless there is some special CVXPY mode or add-on to ...
- 13.1k
4
votes
Accepted
Is not the substitution method supposed to reduce the computation cost?
If you have a long equality constraint $x=\sum_j a_j y_j$ and $x$ appears multiple times in your model, performing the substitution can greatly increase the number of nonzero coefficients in the ...
- 30.3k
4
votes
When is $\max_x\{f(g(x))\} = f(\max_x\{g(x)\})$?
A sufficient condition for $\max_x\{f(g(x))\} = f(\max_x\{g(x)\})$ is that $f$ is a monotone increasing function on the range of $g$. This is also sufficient condition for $\min_x\{f(g(x))\} = f(\...
- 1,347
4
votes
Simplest Quadratic Programming algorithm for teaching
Some really simple but higher-level approaches (no low-level pivoting required) could be:
A simple Sequential Linear Programming algorithm (useful approach for other nonlinear problems and a nice ...
- 2,585
3
votes
Accepted
Min-convex function as constraint
In case $y$ belongs to a finite domain (e.g., if binary), you can split your difficult constraint into multiple simpler (one for each element of the domain) using:
$$
Ax \leq \min_y g(x,y) \quad\...
- 1,347
3
votes
Rational LP, its Rational solution and a minimum precision
Finding $N_{common}$ seems equivalent to finding a column scaling such that the optimal solution is integer-valued. Surely, if you want to do this without knowledge of the optimal solution, you are ...
- 1,347
3
votes
Accepted
Sensitivity analysis for decision vectors in convex programming
Suppose that $F: R^{n} \rightarrow R^{n}$, $F(x)=b$, and $F$ is differentiable at $x$ with non-singular Jacobian $J(x)$. Then to first order, we can use the Jacobian to find a change $\Delta x$ due ...
- 801
3
votes
Does solver take advantage of the problem structure?
Assuming you're talking about the time required to build the model:
Let's time your two approaches first inside an iPython shell via the %%timeit magic command. ...
- 1,477
3
votes
Does solver take advantage of the problem structure?
Probably the content here will be helpful.
Using for-loop outside of the gurobi native construct is slightly faster.
- 3,343
3
votes
How to deal this L0 norm of a vector of L2 or L1 norms in objective?
You can introduce binary variables $\mathbf{z}\in \lbrace 0, 1 \rbrace^n$ (where $n$ is the dimension of $\mathbf{b}$) together with the constraints $b_i \le M_i z_i,$ where $M_i$ is an upper bound ...
- 37.8k
3
votes
Accepted
How to formulate the inequality constraint $\sqrt{x^2+y^2} \leq z$ using gurobipy?
Thanks to the Gurobi staff, this is how to formulate it using the norm constraint:
...
- 391
3
votes
Accepted
Is it possible to make a posynomial concave using a change of variables?
You implicitly assume $c_k > 0$ for all $k$, and $x_i > 0$ for all $i$, and I will do the same. Using the change of variables $\tilde x_i = x_i^M$, such that $x_i = \tilde x_i^{1/M}$, for a big ...
- 1,347
2
votes
How to find the index of the item, the first time appears?
If number is non-binary, say non negative continuous number $c$, introduce a vector of binary $y_i$ initialized to 0.
Have two binary variables $z_1 \ and z_2$
$c - x_i \le My_{i}$
$x_i - c \le M(1-y_{...
- 3,343
2
votes
Accepted
Does the cvxpy replace the max function by MIP formulation under the hood?
No. Firstly you should use cp.maximum instead of cp.max. Secondly, it is converted to a convex programming problem (LP in this ...
- 1,036
2
votes
Does the cvxpy replace the max function by MIP formulation under the hood?
Natively I am not sure since solvers that come do not have MIP capability
Scroll down for list of solvers
And this link to source code max seems to suggest it uses max function.
- 3,343
2
votes
Solver for quadratically constrained mixed-integer linear programs
Gurobi can solve mixed integer quadratically constrained problems provided that your known values in the quadratic constraint form positive semidefinite matrix. You'd need to use Quadratic Constraint ...
- 3,343
2
votes
ADMM diverges on L1 regression
I implemented @batwing's suggestion and it works. Let $\phi_1 \geq 0$, $\phi_2 \geq 0$ be slack variables (which we initialise to $0_N$), the Lagrangian becomes:
$$\mathcal{L}=\alpha^\top 1_N + \...
- 203
2
votes
Accepted
ADMM diverges on L1 regression
ADMM assumes that the constraints being added as penalties are equality constraints. In your reformulation of the l1 objective into inequality constraints i.e., $X\beta - y \leq \alpha$ and $-X \beta +...
- 1,458
Only top scored, non community-wiki answers of a minimum length are eligible