Have you ever seen variables defined like this in JavaScript?

Typically, in JS, variables are defined using var, let, or const.

For example: var abc = 1;

But what if you don't want others to see that you've defined a variable abc? How should you do it?

You can write it like this: this["abc"] = 1;

This also defines a variable abc.

Huh? Can you write it like this? Yes, you can! Don't believe it? Just run a test to find out.

But this still reveals the abc string and shows that the assigned value is the number 1. Is there a way to make it even more obscure?

Yes, you can also use JS-Obfuscator.com to obfuscate this line of code.

After obfuscation, this["abc"] = 1; will become:

this["\u0061\u0062\u0063\u0064"] = 0x3ba17 ^ 0x3ba16;

At this point, if you use console.log(abc) or alert(abc), you will see that the value 1 of the variable abc is output.

However, at a glance, neither the variable abc nor the number 1 is immediately visible in this line of code.

Isn't that interesting?

Have you learned this JS programming trick?

In optimization models, particularly within mixed-integer programming (MIP), the presence of unbounded variables in IMPLIES constraints can lead to a poor big-M relaxation, which affects the efficiency and robustness of the solution process. To address this issue, several strategies can be employed to improve the formulation and avoid numerical instability. One approach is to tighten the bounds on variables whenever possible. Even if variables are theoretically unbounded, practical knowledge about the problem can often provide reasonable bounds. For instance, if a variable represents a physical quantity, such as production capacity, it is unlikely to exceed certain limits. By setting tighter bounds, the big-M value can be reduced, leading to a stronger relaxation[^1]. Another strategy involves the use of indicator constraints, which are supported by some MIP solvers. Indicator constraints allow the direct expression of logical conditions without the need for a big-M formulation. This can significantly improve the model's performance since the solver can handle these constraints more efficiently than traditional big-M approaches[^1]. Additionally, reformulating the problem to avoid IMPLIES constraints altogether can be beneficial. This might involve restructuring the model to use alternative formulations that do not rely on big-M values. For example, using a convex hull formulation for piecewise linear functions can provide a tighter relaxation and better computational performance. When dealing with complex logical conditions, introducing auxiliary variables and constraints can help. These auxiliary variables can be used to decompose the problem into smaller, more manageable parts, which can then be solved more efficiently. This approach can also help in reducing the overall impact of the big-M values on the relaxation quality. Lastly, employing preprocessing techniques can also aid in identifying and mitigating issues related to unbounded variables. Preprocessing can help in tightening the formulation by analyzing the structure of the problem and adjusting the model parameters accordingly. ### Example Code Here is an example of how to implement a tighter bound on a variable in a MIP model using Python and the PuLP library: ```python from pulp import LpProblem, LpVariable, LpMaximize # Create a new problem prob = LpProblem("Example_Problem", LpMaximize) # Define variables with tighter bounds x = LpVariable("x", lowBound=0, upBound=100) # Tightened upper bound y = LpVariable("y", lowBound=0, upBound=50) # Tightened upper bound # Objective function prob += 3 * x + 2 * y # Constraints prob += x + y <= 80 prob += x - y >= 10 # Solve the problem prob.solve() # Print the results for v in prob.variables(): print(v.name, "=", v.varValue) ``` This code snippet demonstrates how to define variables with tighter bounds, which can help in reducing the big-M value and improving the relaxation quality. ###
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