5

Unless you turn off the presolve step, CPLEX will eliminate the variables that are locked at zero during presolve. So the answer to your second question is that presolve may take slightly longer (but the difference will be too small to notice) and the actual branch-and-cut phase will not take any longer than if you had omitted the variables yourself. The ...


4

As a general rule of thumb, I would say that if you have an LP that fits in your memory, then directly applying barrier/simplex is likely to be the best "out-of-the-box" approach. In my experience, decomposition techniques, such as Lagrangean/Benders/Dantzig-Wolfe decomposition, work best (and outperform barrier/simplex) in the following situations ...


3

All equality constraints are active for all feasible solutions. The inequality constraints which are active for a solution are those which are satisfied (hold) with equality at that solution. From the viewpoint of determining how many active constraints are linearly independent, throw all the equality constraints plus the active inequality constraints into ...


2

When you add any number of the original cuts (presumably as constraints, rather than lazy constraints or user cuts) and then restart the solver, CPLEX will go through a presolve step, the results of which may be different from the original presolve due to the presence of the extra constraints. It will then solve the root LP, the solution to which will likely ...


2

Often small LP's are solved as subroutines to solve bigger discrete optimization problems. Example 1: Say you're solving a VRP with time windows and you've a nice heuristic to enumerate good feasible solutions (subgraphs with n-nodes), to figure out an optimal route schedule on the enumerated path, you wanna solve an LP that minimizes the waiting time. This ...


2

As batwing mentioned in a comment, optimal control applications, particularly in robotics, aerospace, and now self-driving cars, involve NLPs to be solved quickly. These NLPs are generally solved using a Sequential Quadratic Programming (SQP) approach, thus involving series of small-changing QPs to be solved very efficiently. For examples of applications, ...


2

I am not familiar with this specific subject but, do you try googling about that? There are many related papers such as: Developing a model for multi-objective optimization of open channels and labyrinth weirs: Theory and application in Isfahan Irrigation Networks Optimization of Irrigation Scheduling Linear Optimization Model for Efficient Use of ...


2

Binary (Boolean) values are integer values. Therefore, optimization problems with boolean constraints are either integer programming or mixed integer programing (MIP). Generally, there is no easy algorithm that is guaranteed to find the optimal solution of MIP problems quickly. My reason to believe that is: I can implement a boolean satisfiability (b-sat) ...


1

Minimize $x^2$ where $1 \le x \le 2$. \begin{aligned} \min_{x} \quad & f(x)\\ \textrm{s.t.} \quad & h_{1}(x) \le 0\\ &h_{2}(x) \le 0 \\ \end{aligned} where \begin{align} f(x) &= x^2 \\ h_{1}(x) &= 1 - x \\ h_{2}(x) &= x - 2 \end{align} KKT conditions: \begin{align} 0 &= \nabla f(x) + \mu_{1}\nabla h_{1}(x) + \mu_{2} \nabla h_{...


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