Optimization: Dynamic Programming vs Kuhn-Tucker Considering the standard utility maximization of representative household which lives forever, one may use dynamic programming and Kuhn-Tucker in case of discrete time. For instance, one would like to solve,

max $\Sigma^∞_tU(C(t),N(t))$
 subject  to $P(t)C(t)+Q(t)B(t)&lt;B(t−1)+W(t)N(t)+D(t)$

where $C(t)$ is consumption, $B$ is the bond, $Q$ is the bond price, $D(t)$ is a dividend, and $N(t)$ is the amount of labor. 

Does the interpretation differ when one use Dynamic programming or Kuhn-Tucker? 
Will it be something like this: In DP all the paths are optimized along t, but in Kuhn-Tucker only the path at time t is optimized. 

If so, how you can make the above statement?
I would say that the main difference stems from the solution method, which results in your statement about all paths versus only the path at time t being true.

Dynamic programming (at least when done numerically) consists of backward induction. One tries to identify the optimal action for all possible values of the state variable in the final period, and then reasons backwards following the state equation. In this way one gets a solution not just for the path we're currently on, but also all other paths. 

Similarly, if one uses the guess-and-verify method to solve the value function of the Bellman equation, the value function one guesses defines the optimal decision for all possible values of the state variable. Thus one gets a solution for all possible paths, including the current one.

Kuhn-Tucker basically works the other way around. One formulates the necessary and sufficient conditions, and solves the resulting difference equation(s) using the initial conditions as starting points.