Maximize the minimum value in a vector

Here's a problem I am trying to solve.

Maximize the minimum value of I:

I = C + R * P (* indicates coordinate-wise product)

where I, C, R and P are n x 1 vectors with the condition that the values of R sum to 1 and all its values be in [0,1]. C and P would be given.

Here is an example with n = 2 (I am writing these vectors transposed so I can type it easily). C is given as [100 120]. P is given as [50 40]

[I1 I2] = [100 120] + [R1 R2] * [50 40]

With n = 2, this breaks down to a system of 2 equations that I was able to solve:

I1 = 100 + (1 - R2) (50)
I2 = 120 + R2 (40)

I set I1 = I2 since this would maximize the minimum value between I1 and I2 (however even this approach would break down under certain conditions).

100 + 50 - 50R2 = 120 + 40R2
150 - 50R2 = 120 + 40R2
30 = 90R2
R2 = 1/3, R1 = 2/3


Is there any way to generally solve this problem? Algorithmic solutions would be helpful as well.

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Originally Posted by daaaaave

Here's a problem I am trying to solve.

Maximize the minimum value of I:

I = C + R * P (* indicates dot product)

where I, C, R and P are n x 1 vectors with the condition that the values of R sum to 1 and all its values be in [0,1]. C and P would be given.

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Are you sure that is what the problem says? R*P is a number while C is a vector. The sum C+ R*P is not defined.

Or do you mean I= (C+ R)*P?

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Here is an example with n = 2 (I am writing these vectors transposed so I can type it easily). C is given as [100 120]. P is given as [50 40]

[I1 I2] = [100 120] + [R1 R2] * [50 40]

With n = 2, this breaks down to a system of 2 equations that I was able to solve:

I1 = 100 + (1 - R2) (50)
I2 = 120 + R2 (40)

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This product is NOT "dot product". You appear to be using [A, B]*[X, Y]= [AX, BY], a "coordinate wise" product.

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I set I1 = I2 since this would maximize the minimum value between I1 and I2 (however even this approach would break down under certain conditions).

100 + 50 - 50R2 = 120 + 40R2
150 - 50R2 = 120 + 40R2
30 = 90R2
R2 = 1/3, R1 = 2/3


Is there any way to generally solve this problem? Algorithmic solutions would be helpful as well.

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Yes, I am sorry. I am talking about a coordinate-wise product, not a dot product.