14 votes

What is a solution?

Great question, @Dirk. People regularly stumble across this, and I believe the notion is not generally agreed upon. Here is how I use it. Main qualifiers for a solution are feasible and optimal. When ...
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12 votes
Accepted

Solvers and saddle points

While iteratively approximately solving the first order Karush-Kuhn-Tucker conditions, many (nonconvex) nonlinear solvers "roll downhill", i.e., enforce descent (for minimization) of the objective ...
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12 votes
Accepted

CPLEX non-convex Quadratic Programming algorithms

The best publicly available CPLEX global QP algorithm description I am aware of is the tutorial presentation by Ed Klotz of IBM at the March 2018 INFORMS Optimization conference. Performance Tuning ...
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10 votes

How to determine different gap rates?

MIP solvers such as CPLEX & Gurobi indicate a gap (in %) between the current best solution and the current best dual bound (which is a lower bound for a minimization problem). In general, the ...
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  • 3,333
9 votes
Accepted

Termination Criteria of Solver in Pyomo

If the IPOPT termination condition is Optimal Solution Found then the returned solution is locally optimal. IPOPT is, by design, not a global solver and therefore ...
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8 votes

What is a solution?

I often encounter a clear difference in the point of view of an operator (business) and a programmer (engineering): From the business POV: if it's not feasible, it's not a solution. Given that an ...
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8 votes
Accepted

Problem solvable $\Rightarrow$ subproblems solvable if feasible region closed, convex?

In your question, you call a problem 'solvable' if there exists an $\hat{x} \in M$ such that \begin{align}c^\top\hat{x} = \inf_{x \in \mathbb{R}^n}&\quad c^\intercal x\\\textrm{s.t.}&\quad x \...
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7 votes

How to determine different gap rates?

Gaps are typically tied to specific models and solution methods. The gap reflects the difference between the best known bound and the objective value of the best solution produced by a particular ...
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7 votes

Recovering primal optimal solutions from dual sub gradient ascent using ergodic primal sequences

In general, nonlinear optimization algorithms implemented in finite precision floating point software don't converge exactly to an optimal solution exactly satisfying the optimality conditions. ...
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6 votes
Accepted

What methods are used to solve multi-objective optimization problem with non-linear objective functions and integer decision variables?

Disclaimer: One might want to look for a reformulation or a special structure to apply mathematical tools to find optimal in the feasible set. I am assuming you're already past the possibility that ...
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6 votes

What is a solution?

Here are two more "dimensions" to the question which have not yet been addressed in any of the other answers, but can be of great significance in practice. Global optimum vs. local optimum: I will ...
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6 votes

What is a solution?

I mostly agree with Marco Lübbecke. I would like to add that "vectors of the right dimension" are sometimes called solution candidates. Also when we refer to an "infeasible solution" we often mean ...
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4 votes
Accepted

Radial unboundedness vs convexity

The notation $C^1$ means $f'$ is continuous (on $\Bbb R$ as the interval is not stated). In general $C^k(a,b]$ means that all of $f',f'',\cdots,f^{(k)}$ are continuous on $(a,b]$. You are correct ...
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4 votes

How to determine different gap rates?

A short little note about computing gaps just appeared in 4OR: Laporte, G., Toth, P. A gap in scientific reporting. 4OR-Q J Oper Res (2021). https://doi.org/10.1007/s10288-021-00483-0
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4 votes

CPLEX non-convex Quadratic Programming algorithms

In addition to the reference of Mark, you can have a look at his technical report: Solving standard quadratic programming by cutting planes. by P. Bonami, A.Lodi, J. Schweiger, A. Tramontani Since ...
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3 votes
Accepted

Does Dijkstra's algorithm find the optimal solution for a weighted and directed shortest paths problem?

I'm assuming the goal here is shortest (least total weight) path. As long as the "problem constraints" affect the graph only to the extent of causing arcs to exist or not exist, and as long ...
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3 votes

CPLEX non-convex Quadratic Programming algorithms

When you call optimize without any options set, the default values will be used, and those are created by the function baronset.
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3 votes

the set of optimal solutions of a linear programming (LP) problem as a mapping of right-hand side

In general, if the LP is bounded, the optimal set $M(b)$ is a face of the feasible set $P = \{ x | Ax = b, x \geq 0\}$ (which is a polyhedral set). In fact, $M$ is a function, but one that maps a ...
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  • 3,333
2 votes

Is Multidisciplinary Design Optimization / Collaborative Optimization used anywhere outside of the Mechanical Engineering context?

As a student I am doing research in this field, I found Wikipedia's explanation very useful. You are right, most of the applications of MDO are in the field of design for aerospace and mechanical ...
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  • 8,340
2 votes

CPLEX non-convex Quadratic Programming algorithms

A non-convex QP is solved to global optimality by generating the McCormick relaxation of the objective and using that relaxation in a branch-and-bound framework. For non-convex MIQPs, we also ...
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1 vote

What is a good approach to deciding which jobs (from a list of HPC jobs) should be ran locally vs. on the cloud given time & cost constraints?

In order to pin down a problem type, you will have to answer a number of questions about the application. How are you going to deal with the cost-time conflict? There are quite a few methods for ...
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