Is there a way to analytically find a stationary point along an arbitrary line in a multivariable quadratic function?Let's say I'm working with a quadratic function with an equation of f ( x ) = 1 2 x T A x − b T x . Now, let's take a direction p and transform the function into one a one dimensional one such that f ′ ( α ) = f ( x 0 + α p ) , where x 0 is some starting point.Is there a way to analytically find an α which minimizes or maximizes f ′ ( α )? What is instead of arbitrary direction p we have p = − ∇ f ( x 0 )?(Yes, this is for an optimization problem)

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Is there a way to analytically find a stationary point along an arbitrary line in a multivariable quadratic function?Let&#039;s say I&#039;m working with a quadratic function with an equation of   f  (      x    )  =      1    2              x        T    A      x    −      b    T        x  . Now, let&#039;s take a direction p and transform the function into one a one dimensional one such that       f    ′    (  α  )  =  f  (            x        0    +  α      p    )  , where             x        0   is some starting point.Is there a way to analytically find an   α which minimizes or maximizes       f    ′    (  α  )? What is instead of arbitrary direction p we have       p    =  −  ∇  f  (            x        0    )?(Yes, this is for an optimization problem)

dobradisamgn · Accepted Answer

Step 1Yes, there is...  f  (      x    )  =      1    2              x        T        A        x    −            b        T        x    ∇  f  (      x    )  =      A        x    −      b    g  (  α  )  =  f  (            x        0    +  α      p    )  g  (  α  )  =      1    2              x        0    T        A              x        0    +  α            p        T        A              x        0    +      1    2        α    2              p        T        A        p    −            b        T              x        0    −  α            b        T        p  Step 2Looking for extremum:       g    ′    (  α  )  =  0      g    ′    (  α  )  =  α            p        T        A        p    +            p        T        A              x        0    −            b        T        p    =  0  α  =  −                              p                T                    A                              x                0            −                        b                T                    p                                      p                T                    A                    p              =  −                              p                T            (              A                              x                0            −              b            )                                p                T                    A                    p              =  −                              p                T            ∇      f      (                        x                0            )                                p                T                    A                    p            And If       p    =  −  ∇  f  (            x        0    )  α  =                              p                T                    p                                      p                T                    A                    p

Is there a way to analytically find a stationary point along an arbitrary line in a multivariable quadratic function? Let's say I'm working with a quadratic function with an equation of f(x)=1/2 x^T Ax-b^T x.

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