# Tag Info

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### 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 ...
• 2,451
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### 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\}&...
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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.
• 22.2k
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### 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 ...
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### 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-...
• 2,719
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### Convexity/Concavity of Average Number of Jobs in M/M/1 Queue?

Your calculations (factoring and simplification) are incorrect. $L$ is neither convex nor concave as a function of $\lambda$ and $\mu$. This can be concluded by examining the eigenvalues of the ...
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### How to solve this convex problem heuristically?

This is a minimum cost flow problem in the bipartite graph $G=(V,A)$ with $V=N_U \cup N_B$. Add a source node and link it to each vertex $v\in N_U$. On each of these arcs, constrain the flow to be in ...
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### How can I express this max-min in CPLEX?

You can model this as a maxmin problem by introducing an auxiliary variable $\theta$: \begin{align} \max&\quad\theta &\\ \text{s.t.}&\quad\theta \leq \sum_{c=1}^C x_{uc}d_{uc} & \...
• 3,056
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### Non-symmetric Positive Definite/Semidefinite Matrix in Quadratic Program

Yes, a real PSD matrix $M$ is a symmetric matrix with $$x^TMx\ge 0$$ for any $x$ (see e.g. https://en.wikipedia.org/wiki/Definite_matrix). However, this is not a real restriction. (We have two ...
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### How to make following constraint a convex one?

The constraint is not convex, and is not transformable to a convex constraint without substantively changing it. The additive linear term $dx$ is irrelevant to convexity. So let's ignore it and look ...
• 10.8k
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### Inverse of weighted sum of positive definite matrices

This is a convex optimization problem, which can easily be formed (and solved), in CVX, among other convex optimization tools. Let $A = x_1I_1 + x_2I_2 + ... + x_nI_n$, where the $I_i$ are m by m ...
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### What does nonconvex multilinear mean?

Non-convex means not convex, which could mean concave but also neither convex nor concave, such as a bilinear term $xy$.
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### Fast way to repeatedly solve many similar LPs/QPs in parallel

The OPTMODEL modeling language in SAS (disclaimer: I work at SAS) supports two features for solving independent optimization (LP or otherwise) problems concurrently: The COFOR loop, which ...
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### Maximize correlation subject to nonconvex correlation constraints

You could add the non-convex constraint $z^Tz = 1$. That would make the objective function and other constraints linear. So this would be a Linear Programming problem, but for a single non-convex ...
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### 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 ...
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### 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.
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• 10.8k

### How can I express this max-min in CPLEX?

Maximize an auxiliary variable $z$ subject to the constraints $z\le \sum_{c=1}^C d_{u,c}x_{u,c}\ \forall u$.
• 29.7k
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### Is $\min \ x^3 \ \mathrm{s.t.}\ x \geq 0$ a convex problem?

An objective function which is convex on only part of its unconstrained domain, but which is convex on the constraint set (i.e.., for any feasible point), can be 'convexified" by modifying the ...
• 10.8k
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### How to convexify log(convex) function?

You are maximizing a convex quadratic (the monotonic log is irrelevant) so the maximum is attained at the border, i.e. either $0$ or $\min(1,\sqrt{1-\text{constant}})$.
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### What is a good way to penalise LP relaxation?

You can't quite create a penalty function to enforce integrality and keep your problem linear, at best it would be quadratic (which could indeed be easier to solve sometimes). The closest thing that ...
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