Now let's consider a real problem or a problem that with some context. Suppose there is a retailer selling products 1 and product 2, let's say the prices are p_1 and p_2. These prices affects the demands for product i the demand is denoted as q_i and that say, it is in this function. p_1 and p_2 affects q_1 in this way, q_1 would be a minus p_1 plus bp_2 and q_2 would be a minus p_2 plus bp_1 according to this general formula. What does that mean? That means when you increase your product one's price, the demand would go down, but when you increase your price for a product 2 whether there is an impact on q_1 depends on the parameter p. Here, the perimeter p is assumed to be within zero and one and that cannot be one. Somehow it's positive. If positive, what does that mean? That means when your p_2 goes up, that's going to increase the demand for product 1. Does that make sense? I think in many cases that makes sense. Suppose product 1 is Coke and the product 2 is Pepsi. Then in that case, if your Pepsi is sold at higher prices, your demand for Coke would be increasing. I hope that makes sense. You may also think product 1 as milk product 2 as, for example water. If the two things are somehow substitutes, then their demands would be affected by the price of the other product. For milk and the water; they don't be that so similar also maybe that p would be small, but not zero, but for Coke and Pepsi, the p would be somewhat higher. Let's assume you know all the numbers, it's just that you now need to do some optimization. Once the thing that you want to do the optimization, you will want to set p_1 and p_2 according to your knowledge on a and b to maximize your total profit and to make our life easier, let's assume there's no production cost. Now we have several problems to do, or we should say we have several steps to solve the whole problem. We want to first ask whether the parameter makes sense? Why we set p to be within zero and one? Then we want to formulate the problem. Check whether it is a convex program. Somehow it would be, otherwise we have no way. Then try to solve the retailers problem and then find an optimal price change according to the change of a and b. Let's try to do this, the first thing is that we want to argue it makes sense to set p to be within zero and one; p somehow cannot be greater than one. Because in that case, the other products price would have an even higher impact then our own price, which I don't think makes sense. For p less than zero somehow its possible, but in that case your two products would be complimentary instead of substituting, so the two products makes more useful things if they are purchased together. Like toothpaste and the brush that you may use to clean your teeth. In that case, the two products should be purchased together. If one thing gets more expensive then the other thing would also get a negative impact like milk and the cereals. That's some cases that we may focus in the future, but today we only want to focus on substitutes. All right? That's one thing and then if we want to formulate the problem, I think it would be this one. We want to choose p_1 and p_2 to maximize the total profit. For product 1 p_1 is the price and that once you determine p_1 and p_2, this is the demand for product 1. For p_2 is the same thing, p_2 would be multiplied by the demand for product 2. You may see that when we choose p_1 and p_2, they are really connected because p_1 and p_2 together determines q_1 and q_2. We want to solve this problem, we want to first check whether it is a convex program. Let's say f of p is the negation of our objective function. Why is that? Because for this function, inside a maximization problem, we want to check whether it is concave, but we don't know how to check concavity. That's make f a negation for this objective function. Then all we need to do is to check whether f is convex. Actually there is also a way regarding the Hessian matrix. To check whether a function is concave, it says that, a function is concave if your Hessian matrix is negative semi-definite, but I don't think we need to bother to introduce that again, because all you need to do, is to get the Functions negation to check whether it is convex. For this one, you may double-check that your Hessian matrix will be looking this way. Two, negative two b, negative two b, and a two. The first leading principle minor is two. That's okay, for the second one is 4 minus 4b square, which is four times one minus b, and the one plus b. Somehow this guy would be always non-negative if we have b within zero and one, I hope that makes sense to have this assumption again, because this exemption actually gives the problem a nice behavior so that it is positive semi definite, so that this function is a convex program. F is convex, negative f would be concave. We are maximizing a concave function and that's why our problem is a convex program. Then now all we need to do is to look for the gradient for our objective function f. The gradient would be zero, and that means we need to have this one and that one. Technically what we need is that our gradient of negative f should be zero, but somehow it has no impact, because whether you have the negation there it's useless if you are talking about equaling to zero. Somehow we need to again solve some two-by-two systems and then very quickly, if you solve these two systems, you will see your p_1 and p_2 must be a over two times one minus p. Maybe you need some hint about how to do this. Well, that's too difficult. For equation one you know, p_1 must be a plus 2bp_2 over two, and then all you need to do is to take this plug in to your second equation, then you are going to get p_2 in one equation, you are going to see that p_2 would be a over two times one minus p and then you plugging that again, you will see p_1. That's how we solve this problem mathematically, but the last thing we want to do is to get some managerial implication from this solution. Our solution is a over two times one minus b. We can see that this quantity increases in a, what does that mean? A is the original demand, a minus p_1 plus p_2 is the demand for q_1. If a is larger somehow, that means our product is more popular and the base demand is higher, so when a product becomes more popular, most people wants to buy it, you will set a higher price. That makes sense. For p, the impact of p is also positive. When p goes larger, what does that mean? Somehow when you increase p_2 by one, that's going to have a better or larger impact on product one and then the other ways also true. When p is larger, it makes sense because the effective demand, q_1 and q_2 tends to be larger and if you have a higher demand, you somehow should instead a higher price. Collectively, that's the whole process given a function or a problem. Now we may formulate non-linear program we may analyze it's convexity, we may solve it through first-order derivatives, second-order derivatives, first-order conditions and then lastly, we may do some interpretations. These help us to understand your problem and to understand your solutions. Pretty much we are down for today. May be you still have some questions in mind, for example, or well, all these problems do not have constraints. How do we deal with non-linear problems with constraints? That will be the main focus for our next lecture. In the next lecture, we will tell you, well, indeed, there are some situations that you do have constraints. There are some ways that we may deal with them that's due within the next time. Okay, that's pretty much I have. Thank you.