I'm working on an optimization problem involving a single decision variable p1, subject to four inequality constraints:

• Two upper bound constraints:

  p1 < b1,p1 < b2

• Two lower bound constraints:

  c1 ≤ p1,c2 ≤ p1

Effectively, the feasible region for p1 is:

max⁡(c1,c2)  ≤  p1  ≤  min⁡(b1,b2)

I have already formulated the Karush-Kuhn-Tucker (KKT) conditions for this setup, and now I'm trying to determine:

1. The optimal value p1∗​

2. The corresponding feasibility conditions

3. A case-wise breakdown depending on which constraints are active or inactive

Sheet:  Q_P1_Optimum_condition.mw

How can I rearrange the inequality to isolate p1​ on one side?
I'm currently using the 'isolate' function, but it's returning expressions with the signum function.

Q_isolate.mw

Given two optimality conditions defined as inequalities — a lower bound condition and an upper bound condition — under the assumptions that all parameters are positive and satisfy the relationships w>Pu>Ce>U[0]>0, and w>Pu>Ce,   and upsilon,varphi, tau0, eta are between (0,1), where 1>=varphi>=tau0>=0. and upsilon>eta

Under what conditions on the parameters does the lower bound condition become violated while the upper bound condition is also violated?

Detailed question in sheet attached: Q_Condition.mw

[ see also Q_Const.mw ]

I am working on a script (attached sheet) where I use a FOR loop to iterate over different values of varepsilon. Within each iteration, I perform the following steps:

1. I optimize the function R_out and obtain a result, denoted as Pc

2. I then substitute Pc into another function, L_out and optimize it to find the values of p1,p2, and the corresponding function value.

I follow a similar procedure for a second case as well.

However, I'm encountering an issue: I'm unable to successfully substitute the result from R_out into L_out within the loop. This is causing an error in execution.

Finally, I intend to generate a plot with varepsilon  on the x-axis and certain result variables on the y-axis in a single plot (as indicated at the end of the sheet).

Sheet: Question_New.mw

Can someone help me:

1. Fix the substitution error in the loop, and

2. Provide the correct syntax for generating the desired plot?

I'm optimizing a function with constraints, meaning the decision variable has both a lower and an upper bound. When I use the  (MAX) syntax in Maple, it returns the lower bound as the optimal value (Pc) along with the corresponding function value. However, when I plot the graph, it shows that the function actually reaches its maximum at the upper bound. What could be causing this discrepancy

Sheet:Q_result_1.mw

Suppose we have two equations, and all parameters involved are assumed to be positive. Is there a systematic way or syntax to determine whether C1>C2​ or vice versa? Additionally, can we derive specific conditions under which C1>C2​ holds

Sheet: Q_greater.mw

I need to create a slider plot for A10, A11, and A12 by varying the parameters theta, Pu, and a.
I have a syntax ready — could you suggest modifications to make it work correctly and generate the plot?

Additionally, is it possible to compute the values of A13 and A14 by substituting the obtained A10, A11, and A12 values for each combination of theta, Pu, and a from the slider plot?

Sheet attached: Slider_Q.mw

I have two conditions, A and B, which must both be greater than or equal to zero. However, when I try to solve the problem, I encounter an error.

Could you please help me resolve this?

Note: All the parameters, whether specified or not, are positive and greater than zero.
File : Q3.mw

I want both Pc and r(Pc) to be greater than or equal to zero. The only constraint is that all parameters for which I’ve provided data must remain positive. Can we identify the key parameters that significantly affect Pc? Also, what condition ensures that Pc ≥ 0? Ideally, I’d like Pc to be less than or equal to Pu. Could you suggest what changes in the numerical values I should make to ensure Pc becomes a positive value?

Attaching file: Q2.mw

Can we include a graph that shows how Pc changes with respect to variations in the most sensitive parameters?

I’m trying to simplify the results I obtained from two scenarios. In the first scenario, I have results labeled g1,g2,g3​, and in the second scenario, I have g_1, g_2, g_3​. The issue is that the expressions are quite lengthy, and I want to shorten them by identifying and substituting common sub-expressions.

Is there a method or syntax that can help me automatically detect repeated terms and substitute them with a variable to improve readability?

For example:

Let’s say:

x=2f+3d+4(d+h+k)j+f

y=1(d+h+k)+hf+4d

Here, the expression d+h+k appears in both x and y, so I can define a new variable:

A=d+h+k

and substitute it into the equations to make them more concise and readable.

Attaching the sheet: Difference_two_model.mw

I have two equations on either side of an inequality that contain like terms such as Am and Ce. Could you simplify the expressions by mathematically eliminating these common terms from both sides? For example, if we have an equation like x+y⋅d+hx=g⋅f+hx , it simplifies to x+y⋅d=g⋅f.

Additionally, please solve for Cv and Ce.

Note: All terms are positive except R0er and R0m​.

I am attaching the relevant sheet for reference. Q_12.mw

Could we create a plot with tau0 varying from 0.1 to 0.6 on the x-axis and profit on the y-axis displaying Rprof, Mprof, Tprof, T_Cprof all on the same graph?

Sheet attached : trial_question.mw

I would like the graph to display the optimal point clearly marked with a red dot. Additionally, the optimal point should be labeled for easy identification. what is the syntax for it.
The  sheet is attached below :
trial_question.mw

ContoursWithLabels(............................, labels = ["delta0", "s2"]);

how to make delta₀  and s₂ ? and delta should be symbol not word?

The maple worksheet shows an incorrect evaluation of the integral in (1) which is a standard integral representation of a Bessel function.  Equations (2)-(5) along with the graph show the incorrectness of the evaluation.  What is going on?

Bessel.mw