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(...
Henrik Alsing Friberg's user avatar
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. ...
prubin's user avatar
  • 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).
Enrique Gabriel Baquela's user avatar
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{...
RobPratt's user avatar
  • 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 ...
Sutanu Majumdar's user avatar
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 + ...
xd y's user avatar
  • 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/ ...
Oscar Dowson's user avatar
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, ...
Henrik Alsing Friberg's user avatar
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 ...
Sutanu Majumdar's user avatar
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 "...
Mark L. Stone's user avatar
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 ...
batwing's user avatar
  • 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 ...
Henrik Alsing Friberg's user avatar
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 ...
Mark L. Stone's user avatar
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 ...
RobPratt's user avatar
  • 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(\...
Henrik Alsing Friberg's user avatar
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 ...
Erwin Kalvelagen's user avatar
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\...
Henrik Alsing Friberg's user avatar
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 ...
Henrik Alsing Friberg's user avatar
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 ...
Brian Borchers's user avatar
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. ...
joni's user avatar
  • 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.
Sutanu Majumdar's user avatar
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 ...
prubin's user avatar
  • 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: ...
Hussein Sharadga's user avatar
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 ...
Henrik Alsing Friberg's user avatar
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_{...
Sutanu Majumdar's user avatar
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 ...
xd y's user avatar
  • 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.
Sutanu Majumdar's user avatar
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 ...
Sutanu Majumdar's user avatar
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 + \...
Carol Eisen's user avatar
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 +...
batwing's user avatar
  • 1,458

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