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UPDATED FALL 2018 2 Example I.3. (Rate and power over a 3-user wireless system) Consider a wireless device that transmits to three different users over orthogonal links. The device must choose a power vector (p1 , p2 , p3 ) ∈ R3 that satisfies the following constraints: p1 + p2 + p3 pi ≤ pmax (1) > 0 ∀i ∈ {1, 2, 3} (2) where pmax is a positive real number that constrains the sum power usage. For each i ∈ {1, 2, 3}, let µi (pi ) = log(1 + γi pi ) be the transmission rate achieved over link i as a function of the power variable pi , where γi is some known attenuation coefficient for link i. Define M as the set of all (p1 , p2 , p3 ) ∈ R3 that satisfy the constraints (1)-(2). Define x(p1 , p2 , p3 ) = p1 + p2 + p3 y(p1 , p2 , p3 ) = −[µ1 (p1 ) + µ2 (p2 ) + µ3 (p3 )] Thus, x(p1 , p2 , p3 ) represents the sum power used, while y(p1 , p2 , p3 ) is −1 times the sum rate over all three links. The goal is to choose (p1 , p2 , p3 ) ∈ M to keep both x(p1 , p2 , p3 ) and y(p1 , p2 , p3 ) small. Example I.4. (Network utility maximization) Consider the same 3-link wireless system as Example I.3. However, suppose we do not care about power expenditure. Rather, we care about: • Maximizing the sum rate µ1 (p1 ) + µ2 (p2 ) + µ3 (p3 ). • Maximizing the proportionally fair utility metric log(µ1 (p1 )) + log(µ2 (p2 )) + log(µ3 (p3 )). This is a commonly used notion of fairness for rate allocation over multiple users.1 Again let M be the set of all vectors (p1 , p2 , p3 ) ∈ R3 that satisfy (1)-(2). Define x(p1 , p2 , p3 ) = −[µ1 (p1 ) + µ2 (p2 ) + µ3 (p3 )] y(p1 , p2 , p3 ) = −[log(µ1 (p1 )) + log(µ2 (p2 )) + log(µ3 (p3 ))] so that x(p1 , p2 , p3 ) is −1 times the sum rate, and y(p1 , p2 , p3 ) is −1 times the proportionally fair utility metric. The goal is to choose (p1 , p2 , p3 ) ∈ M to minimize both x(p1 , p2 , p3 ) and y(p1 , p2 , p3 ). Example I.1 emphasizes that the set M can have any size and structure that we want, and its elements can be any type of object that we want (in that example, M is a finite set of possible routes). Examples I.2-I.4 show that the set M can be an infinite set of vectors (p1 , p2 , p3 ). Examples I.2-I.4 also show how a bicriteria optimization problem that seeks to maximize one objective while minimizing another, or that seeks to maximize both objectives, can be transformed into a bicriteria minimization problem by multiplying the appropriate objectives by −1. Hence, without loss of generality, it suffices to assume the system controller wants both components of the vector of objectives (x, y) to be small. A. Pareto optimality Define A as the set of all (x, y) vectors in R2 that are achievable via system modes m ∈ M: A = {(x(m), y(m)) ∈ R2 : m ∈ M} Every (x, y) pair in A is a feasible operating point. Once the set A is known, system optimality can be understood in terms of selecting a desirable 2-dimensional vector (x, y) in the set A. With this approach, the study of optimality does not require knowledge of the physical tasks the system must perform for each mode of operation in M. This is useful because it allows many different types of problems to be treated with a common mathematical framework. The set A can have an arbitrary structure. It can be finite, infinite, closed, open, neither closed nor open, and so on. Assume the system controller wants to find an operating point (x, y) ∈ A for which both x and y are small. Definition I.1. A vector (x, y) ∈ A is preferred over (or dominates) another vector (w, z) ∈ A, written (x, y) ≺ (w, z), if the following two inequalities hold • x≤w • y ≤z and if at least one of the inequalities is strict (so that either x < w or y < z). Definition I.2. A vector (x∗ , y ∗ ) ∈ A is Pareto optimal if there is no vector (x, y) ∈ A that satisfies (x, y) ≺ (x∗ , y ∗ ). A set can have many Pareto optimal points. An example set A and its Pareto optimal points are shown in Fig. 1. For each vector (a, b) ∈ R2 , define S(a, b) as the set of all points (x, y) that satisfy x ≤ a and y ≤ b: S(a, b) = {(x, y) ∈ R2 : x ≤ a, y ≤ b} 1 See [6] for a development of proportionally fair utility and its relation to the log(µ) function. The constraints (2) avoid the singularity of the log(µ) function at 0, so that log(µ1 (p1 )) + log(µ2 (p2 )) + log(µ3 (p3 )) is indeed a real number whenever (p1 , p2 , p3 ) satisfies (1)-(2). An alternative is to use constraints pi ≥ 0 (which allow zero power in some channels), but to modify the utility function from log(µ) to (1/b) log(1 + bµ) for some constant b > 0.

UPDATED FALL 2018 3 Pictorially, the set S(a, b) is an infinite square in the 2-dimesional plane with upper-right vertex at (a, b) (see Fig. 1). If (a, b) is a point in A, any other vector in A that is preferred over (a, b) must lie in the set S(a, b). If there are no points in A ∩ S(a, b) other than (a, b) itself, then (a, b) is Pareto optimal. Set$A (a,b)$ S(a,b)$ Pareto$op*mal$$ points$ Fig. 1. An example set A (in orange) that contains an irregular-shaped connected component and 7 additional isolated points. The Pareto optimal points on the connected component are colored in green, and the two Pareto optimal isolated points are circled. The rectangle set S(a, b) is illustrated for a particular Pareto optimal point (a, b). Note that (a, b) is Pareto optimal because S(a, b) intersects A only at the point (a, b). B. Degenerate cases and the compact assumption In some cases the set A will have no Pareto optimal points. For example, suppose A is the entire set R2 . If we choose any point (x, y) ∈ R2 , there is always another point (x − 1, y) ∈ R2 that is preferred. Further, it can be shown that if A is an open subset of R2 , then it has no Pareto optimal points (see Exercise IX-A.5). To avoid these degenerate situations, it is often useful to impose the further condition that the set A is both closed and bounded. A closed and bounded subset of RN is called a compact set. If A is a finite set then it is necessarily compact. It can be shown that if A is a nonempty compact set, then: 1) It has Pareto optimal points. 2) For every point (a, b) ∈ A that is not Pareto optimal, there is a Pareto optimal point that is preferred over (a, b). See Exercise IX-A.11 for a proof of the above two claims. Therefore, when A is compact, we can restrict attention to choosing an operating point (x, y) that is Pareto optimal. II. O PTIMIZATION WITH ONE CONSTRAINT 2 Let A ⊆ R be a set of all feasible (x, y) operating points. Assume the system controller wants to make both components of the vector (x, y) small. One way to approach this problem is to minimize y subject to the constraint x ≤ c, where c is a given real number. To this end, fix a constant c ∈ R and consider the following constrained optimization problem: Minimize: y (3) Subject to: x≤c (4) (x, y) ∈ A (5) The variables x and y are the optimization variables in the above problem, while the constant c is assumed to be a given and fixed parameter. The above problem is feasible if there exists an (x, y) ∈ R2 that satisfies both constraints (4)-(5). Definition II.1. A point (x∗ , y ∗ ) is a solution to the optimization problem (3)-(5) if the following two conditions hold: ∗ ∗ • (x , y ) satisfies both constraints (4)-(5). ∗ • y ≤ y for all points (x, y) that satisfy (4)-(5). It is possible for the problem (3)-(5) to have more than one optimal solution. It is also possible to have no optimal solution, even if the problem is feasible. This happens when there is an infinite sequence of points {(xn , yn )}∞ n=1 that satisfy the constraints (4)-(5) with strictly decreasing values of yn , but for which the limiting value of yn cannot be achieved (see Exercise IX-B.1). This can only happen if the set A is not compact. On the other hand, it can be shown that if A is a compact set, then the problem (3)-(5) has an optimal solution whenever it is feasible.

UPDATED FALL 2018 4 A. The tradeoff function The problem (3)-(5) uses a parameter c in the inequality constraint (4). If the problem (3)-(5) is feasible for some given parameter c, then it is also feasible for every parameter c0 that satisfies c0 ≥ c. Thus, the set of all values c for which the problem is feasible forms an interval of the real number line of the form either (cmin , ∞) or [cmin , ∞). Call this set the feasibility interval. The value cmin is the infimum of the set of all real numbers in the feasibility interval. For each c in the feasibility interval, define ψ(c) as the infimum value of the objective function in problem (3)-(5) with parameter c. In particular, if (x∗ , y ∗ ) is an optimal solution to (3)-(5) with parameter c, then ψ(c) = y ∗ . If A is a compact set, it can be shown that the feasibility interval has the form [cmin , ∞) and that problem (3)-(5) has an optimal solution for all c ∈ [cmin , ∞). The function ψ(c) is called the tradeoff function. The tradeoff function establishes the tradeoffs associated with choosing larger or smaller values of the constraint c. Intuitively, it is clear that increasing the value of c imposes less stringent constraints on the problem, which allows for improved values of ψ(c). This is formalized in the next lemma. ψ(c)" c1" cmin" c2" c3" c" Fig. 2. The set A from Fig. 1 with its (non-increasing) tradeoff function ψ(c) drawn in green. Note that ψ(c) is discontinuous at points c1 , c2 , c3 . Lemma II.1. The tradeoff function ψ(c) is non-increasing over the feasibility interval. Proof. For simplicity assume A is compact. Consider two values c1 and c2 in the interval [cmin , ∞), and assume c1 ≤ c2 . We want to show that ψ(c1 ) ≥ ψ(c2 ). Let (x∗1 , y1∗ ) and (x∗2 , y2∗ ) be optimal solutions of (3)-(5) corresponding to parameters c = c1 and c = c2 , respectively. Then: y1∗ = ψ(c1 ) y2∗ = ψ(c2 ) By definition of (x∗2 , y2∗ ) being optimal for the problem with parameter c = c2 , we know that for any vector (x, y) ∈ A that satisfies x ≤ c2 , we have: y2∗ ≤ y (6) On the other hand, we know (x∗1 , y1∗ ) is a point in A that satisfies x∗1 ≤ c1 ≤ c2 , so (6) gives: y2∗ ≤ y1∗ Substituting y1∗ = ψ(c1 ) and y2∗ = ψ(c2 ) gives the result. Note that the tradeoff function ψ(c) is not necessarily continuous (see Fig. 2). It can be shown that it is continuous when the set A is compact and has a convexity property.2 Convexity is defined in Section IV. The tradeoff curve is defined as the set of all points (c, ψ(c)) for c in the feasibility interval. Exercise IX-A.8 shows that every Pareto optimal point (x(p) , y (p) ) of A is a point on the tradeoff curve, so that ψ(x(p) ) = y (p) . 2 In particular, ψ(c) is both continuous and convex over c ∈ [c min , ∞) whenever A is compact and convex. Definitions of convex set and convex function are provided in Section IV.

UPDATED FALL 2018 5 B. Lagrange multipliers for optimization over (x, y) ∈ A The constrained optimization problem (3)-(5) may be difficult to solve because of the inequality constraint (4). Consider the following related problem, defined in terms of a real number µ ≥ 0: Minimize: y + µx (7) Subject to: (x, y) ∈ A (8) The problem (7)-(8) is called the unconstrained optimization problem because it has no inequality constraint. Of course, it still has the set constraint (8). The constant µ is called a Lagrange multiplier. It acts as a weight that determines the relative importance of making the x component small when minimizing the objective function (7). Note that if (x∗ , y ∗ ) is a solution to the unconstrained optimization problem (7)-(8) for a particular value µ, then: y ∗ + µx∗ ≤ y + µx for all (x, y) ∈ A (9) In particular, all points of the set A are on or above the line consisting of points (x, y) that satisfy y + µx = y ∗ + µx∗ . This line has slope −µ and touches the set A at the point (x∗ , y ∗ ) (see Fig. 3). slope&=&/μ& (x*,&y*)& slope&=&/μ& A A H (x1*,&y1*)& (a)&& (b)&& (x2*,&y2*)& Fig. 3. (a) An example set A and multiplier µ. The point (x∗ , y ∗ ) is the single minimizer of y + µx over (x, y) ∈ A. (b) The same set A with a different multiplier µ. Points (x∗1 , y1∗ ) and (x∗2 , y2∗ ) both minimize y + µx over (x, y) ∈ A. The set H shown in the figure contains “hidden Pareto optimal points” that cannot be found via global minimization of x + µy over (x, y) ∈ A, regardless of the value of µ. The next theorem shows that if a point (x∗ , y ∗ ) solves the unconstrained problem (7)-(8) for a particular parameter µ ≥ 0, then it must also solve the constrained problem (3)-(5) for a particular choice of the c value, namely, c = x∗ . Theorem II.1. If (x∗ , y ∗ ) solves the unconstrained problem (7)-(8), then: a) If µ ≥ 0, then (x∗ , y ∗ ) also solves the following optimization problem (where (x, y) are the optimization variables and ∗ x is treated as a given constant): Minimize: y (10) Subject to: x ≤ x∗ (11) (x, y) ∈ A (12) b) If µ > 0, then (x∗ , y ∗ ) is Pareto optimal in A. Proof. To prove part (a), suppose (x∗ , y ∗ ) solves the unconstrained problem (7)-(8). Then (x∗ , y ∗ ) also satisfies the constraints of problem (10)-(12). Indeed, the constraint (11) is trivially satisfied by the vector (x∗ , y ∗ ) because the first variable of this vector is less than or equal to x∗ (that is, x∗ ≤ x∗ is a trivially true inequality). Further, vector (x∗ , y ∗ ) also satisfies (12) because this constraint is the same as (8). Next, we want to show that (x∗ , y ∗ ) is a solution to (10)-(12). Let (x, y) be any other vector that satisfies (11)-(12). It suffices to show that y ∗ ≤ y. Since (x, y) ∈ A we have from (9): y ∗ + µx∗ ≤ y + µx ≤ y + µx∗ where the final inequality follows from (11) together with the fact that µ ≥ 0. Simplifying the above inequality gives y ∗ ≤ y. This proves part (a). The proof of part (b) is left as an exercise (see Exercises IX-A.9 and IX-A.10).

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