# Tag Info

Accepted

### Why does some solvers can only solve conic optimization problems?

The fundamental algorithm employed by Mosek and SeDuMi is a primal-dual interior-point algorithm based on the work of Nesterov and Todd (NT). See for instance my paper and the references therein. This ...
• 2,816

### What is the best open source solver for large scale LP optimization in pyomo?

There is a new open source solver that looks quite promising, HiGHS: https://www.maths.ed.ac.uk/hall/HiGHS/ But as pointed out by others, for mixed-integer programming problems, at the moment, open-...
• 3,148
Accepted

### How to check for convexity of the inequality constraint $−x^2+y−1\ge0$ for a minimization objective function?

I think you are trying to use the following property: $\{x:g(x) \le 0\}$ is convex if $g$ is convex. Note the direction of the inequality. Notice that \begin{align}\{(x,y): -x^2+y-1 \ge 0\}&...
Accepted

Rather than solving this directly as MIQCQP, you might consider linearizing the products $y_{ij} x_{ij}$, as shown here, yielding instead an MILP problem.
• 27.3k
Accepted

### Express equality constraint involving exponentials cones

Q: "How do i write $\text{exp}(a) = b$ using cone programming?" A You don't. $\text{exp}(a) = b$ is a nonlinear equality constraint, and is therefore non-convex. $\text{exp}(a) \le b$ is ...
• 12.2k
Accepted

• 27.3k
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{...
• 27.3k

### 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).

### Difference between exploration and exploitation in Simulated Annealing algorithm

I personally see it as follows. In simulated annealing the likelihood of choosing a solution from the neighborhood is quite high at the beginning. This phase could be regarded as exploration as the ...
• 1,658

### Convex Maximization with Linear Constraints

The location-inventory problem by Shen, et al. and Daskin, et al. has a concave minimization objective. It's related to economies of scale (which you list in your PS 2) but not exactly the same.
• 12.9k

• 1,016

### Disciplined convex programming representation of $x\sqrt{1-x}$

This is possible purely under DCP. As you are interested in the interval $[0,1]$, rewrite your function as $$x\sqrt{1-x}=\exp\left(\ln x+\frac12\ln(1-x)\right),\quad x\in[0,1].$$ Then the following ...
• 5,271

### Disciplined convex programming representation of $x\sqrt{1-x}$

I don't think you can represent this as a concave function in the (cvxpy-)DCP sense: import cvxpy as cp x=cp.Variable() a=x*cp.sqrt(1-x) a.curvature 'UNKNOWN' ...
• 543
Accepted

### Approximation methods for a mixed integer convex optimization problem

Mosek 9.x can natively solve mixed-integer exponential cone problems. Formulate the problem in YALMIP, specifying the binary variables as binvar, and Mosek as the solver. YALMIP will call Mosek to ...
• 12.2k