# Sensitivity analysis for decision vectors in convex programming

Can we perform sensitivity analysis on the decision variables for the perturbed right-hand side of the constraints in a convex/nonlinear program? I know a basic result regarding the sensitivity of the optimal solution but are there any techniques/references to do the same for the decision variables?

Suppose that $$F: R^{n} \rightarrow R^{n}$$, $$F(x)=b$$, and $$F$$ is differentiable at $$x$$ with non-singular Jacobian $$J(x)$$. Then to first order, we can use the Jacobian to find a change $$\Delta x$$ due to a small change $$\Delta b$$.

$$F(x+\Delta x)=b+\Delta b$$.

$$F(x)+J(x)\Delta x=b+\Delta b$$.

$$\Delta x=J(x)^{-1} \Delta b$$.

This is exact only in the limit as $$\Delta b$$ goes to 0, but it tells you that the partial derivatives of $$x_{i}$$ with respect to $$b_{j}$$ are given by

$$\frac{\partial x_{i} }{\partial b_{j}}=(J(x)^{-1})_{i,j}$$.

Now suppose that you've got a smooth optimization problem

$$\min f(x)$$

subject to

$$h_{i}(x)=b_{i}$$, $$i=1, 2, \ldots, m$$

and that you have found an optimal solution $$x^{*}$$ and also solved the KKT conditions. The KKT conditions are of the form

$$K(x^{*},\lambda^{*})=0$$

including the specific constraint equations

$$h_{i}(x^{*})-b_{i}=0$$, $$i=1, 2, \ldots, m$$

and the Lagrange multiplier constraints

$$\nabla f(x^{*}) + \sum_{i=1}^{m} \lambda_{i}^{*} \nabla h_{i}(x^{*})=0$$.

You can now find the Jacobian of $$K(x^{*},\lambda^{*})$$ and invert it to get the sensitivity of $$x^{*}$$ with respect to changes in $$b_{i}$$.

Note that in order to do this you'll have to look at the second derivatives of $$f(x)$$ and the $$h_{i}(x)$$ since $$K$$ already involves the first derivatives.

You can easily extend this to problems with inequality constraints.

A good book that discusses this is:

Bonnans, J. Frédéric, and Alexander Shapiro. Perturbation Analysis of Optimization Problems. Springer, 2013.

• thank you so much! A follow-up question: is it possible to do the same thing but instead of changing the right-hand side of the constraint, we change some parameters in the objective function? (we still look at the effect on the optimal decision vector.
– T_k
Apr 6 at 23:02
• Consider how those changes in the objective function change $\nabla f(x)$ in the KKT conditions. Apr 7 at 3:55