Consider the following maximization problem. max x ∈ R n x T P x a i ≤ x i ≤ b i , ∀ i ∈ Z [ 1 , n ] , where x i is the i-th scalar component of vector x and P &gt; 0 is positive definite. As the objective is a convex function, the maximizer should be where all constraints are active, i.e. at an extreme point of the domain of the form x i ⋆ = a i or x i ⋆ = b i . Is the global maximizer the one of such points such that its norm x ⋆ T x ⋆ is maximal?

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Consider the following maximization problem.                                            max                      x            ∈                                          R                            n                                                x          T                P        x                                            a          i                ≤                  x          i                ≤                  b          i                ,                 ∀        i        ∈                              Z                                [            1            ,            n            ]                                  ,            where       x    i   is the   i-th scalar component of vector   x and   P  &amp;gt;  0 is positive definite. As the objective is a convex function, the maximizer should be where all constraints are active, i.e. at an extreme point of the domain of the form       x    i    ⋆    =      a    i   or       x    i    ⋆    =      b    i  . Is the global maximizer the one of such points such that its norm                     x        ⋆              T        x    ⋆   is maximal?

Karissa Macdonald · Accepted Answer

Maximizing                     x        →              T              x      →       is not generally equivalent to maximizing                     x        →              T    P            x      →       . As a simple counterexample, take this choice of parameters for             x      →        ∈            R        2  .  P  =      [                            2                          −          1                                      −          1                          2                      ]    −  1  ≤      x    1    ≤  2  −  1  ≤      x    2    ≤  2We see that                     x        →              T              x      →        is maximized at             x      →        =      [                            2                                      2                      ]  , while                     x        →              T    P            x      →        is maximized at             x      →        =      [                            2                                      −          1                      ]    ,      [                            −          1                                      2                      ]  .

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