Bilevel Portfolio Optimization: A Global Optimization Approach World Congress on Global Optimization, 2015, Centre of Applied Optimization, University of Florida, Gainesville, FL, USA. Nivedita Haldar, IIM CAlcutta

PORTFOLIO OPTIMIZATION A portfolio consists of various amounts of money invested across different assets The goal of portfolio optimization : to find the optimal values of the fractions of the wealth invested in each asset which will yield a larger value of the expected portfolio return and smaller value of risk But these two goals tend to conflict each other So the investor would try for an efficient portfolio defined by Markowitz’s Mean-Variance Model World Congress on Global Optimization, 2015, Centre of Applied Optimization, University of Florida, Gainesville, FL, USA.

PORTFOLIO OPTIMIZATION World Congress on Global Optimization, 2015, Centre of Applied Optimization, University of Florida, Gainesville, FL, USA.

EFFICIENT PORTFOLIOS A portfolio is variance-efficient if for a fixed return Rp, it has the minimum risk (variance) A portfolio is expected return-efficient if for a fixed variance Vp, it yields the maximum expected return World Congress on Global Optimization, 2015, Centre of Applied Optimization, University of Florida, Gainesville, FL, USA.

EFFICIENT PORTFOLIOS A portfolio is parametric-efficient if for some non-negative parameter ρ, it seeks highest return with lowest risk; i.e., where ρ = 0 indicates minimum-variance portfolio with the investor being totally risk-averse; with the increase of ρ, the investor becomes more risk-taker and he gets higher expected returns at higher risks World Congress on Global Optimization, 2015, Centre of Applied Optimization, University of Florida, Gainesville, FL, USA.

FINDING EFFICIENT PORTFOLIOS World Congress on Global Optimization, 2015, Centre of Applied Optimization, University of Florida, Gainesville, FL, USA.

FINDING EFFICIENT PORTFOLIOS World Congress on Global Optimization, 2015, Centre of Applied Optimization, University of Florida, Gainesville, FL, USA.

FINDING EFFICIENT PORTFOLIOS World Congress on Global Optimization, 2015, Centre of Applied Optimization, University of Florida, Gainesville, FL, USA.

FINDING EFFICIENT PORTFOLIOS World Congress on Global Optimization, 2015, Centre of Applied Optimization, University of Florida, Gainesville, FL, USA.

FINDING EFFICIENT PORTFOLIOS World Congress on Global Optimization, 2015, Centre of Applied Optimization, University of Florida, Gainesville, FL, USA.

EFFICIENT FRONTIER For different variances we shall get different expected return-efficient portfolios For different expected returns we shall get different variance-efficient portfolios The collection of efficient portfolios thus obtained forms the ‘efficient frontier’ of the portfolio universe If we plot risks measured in variances or standard deviations (V, say) along the horizontal axis and expected returns (R, say) along the vertical axis against different parametric-efficient portfolios for different ρ we get an efficient frontier curve World Congress on Global Optimization, 2015, Centre of Applied Optimization, University of Florida, Gainesville, FL, USA.

World Congress on Global Optimization, 2015, Centre of Applied Optimization, University of Florida, Gainesville, FL, USA.

AN EXAMPLE World Congress on Global Optimization, 2015, Centre of Applied Optimization, University of Florida, Gainesville, FL, USA.

PARAMETRIC-EFFICIENT PORTFOLIO World Congress on Global Optimization, 2015, Centre of Applied Optimization, University of Florida, Gainesville, FL, USA.

RESULT: ρ R V y1 y2 y3 F 1.22 0.001 0.62 0.35 0.03 0.05 1.23 0.002 0.7 0.3 -0.061 0.1 1.24 0.003 0.77 0.23 -0.123 0.15 1.26 0.006 0.83 0.17 -0.186 0.2 1.27 0.01 0.9 -0.249 0.25 1.28 0.015 0.96 0.04 -0.313 -0.377 0.018 1 -0.439 0.4 -0.503 World Congress on Global Optimization, 2015, Centre of Applied Optimization, University of Florida, Gainesville, FL, USA.

World Congress on Global Optimization, 2015, Centre of Applied Optimization, University of Florida, Gainesville, FL, USA.

BILEVEL PROGRAMMING It is a nested optimization technique for solving decentralized planning problems involves hierarchical decision-making in which the upper level decision maker (called the leader/ superior/ top planner) influences the lower level decision maker (called the follower/ inferior/ bottom planner)and vice versa Sherali et al. [1988] have shown that bilevel programming is a method to solve a static Stackelberg game (Stackelberg [1952]) in oligopoly market World Congress on Global Optimization, 2015, Centre of Applied Optimization, University of Florida, Gainesville, FL, USA.

CHARACTERISTICS OF BILEVEL PROGRAMMING (Bialas and Karwan [1984]) The decision making units are interactive but exist in a hierarchical structure Each unit independently controls a set of decision variables disjoint from the other The decision making is sequential from the upper level to the lower level; the follower executes its policies after, and in view of, the decisions of the leader The follower is always rational while responding to the decision made by the leader Each decision making unit optimizes its own objective function independently of other unit but may get influenced by the actions and reactions of other decision making unit, and thereby improves its own objective function The decision making units do not cooperate with each other, the objective functions are often conflicting in nature World Congress on Global Optimization, 2015, Centre of Applied Optimization, University of Florida, Gainesville, FL, USA.

BILEVEL PROGRAMMING World Congress on Global Optimization, 2015, Centre of Applied Optimization, University of Florida, Gainesville, FL, USA.

BILEVEL PORTFOLIO OPTIMIZATION Generally investors invest money through broker houses by paying some brokerage charges The broker house would try to maximize the total brokerage charge per dollar of total investment the investor may not be willing to pay more than a certain percentage of the total investment as the total brokerage charge the broker house and the investor would involve in a Stackelberg game with conflicting interests where the broker house, as a leader, will set brokerage charges first and then the investor, as a follower, will choose the fraction of investments in accordance with the given brokerage charges to get an efficient portfolio World Congress on Global Optimization, 2015, Centre of Applied Optimization, University of Florida, Gainesville, FL, USA.

BILEVEL PORTFOLIO OPTIMIZATION World Congress on Global Optimization, 2015, Centre of Applied Optimization, University of Florida, Gainesville, FL, USA.

BILEVEL PORTFOLIO OPTIMIZATION World Congress on Global Optimization, 2015, Centre of Applied Optimization, University of Florida, Gainesville, FL, USA.

BILEVEL PORTFOLIO OPTIMIZATION World Congress on Global Optimization, 2015, Centre of Applied Optimization, University of Florida, Gainesville, FL, USA.

BILEVEL PORTFOLIO OPTIMIZATION World Congress on Global Optimization, 2015, Centre of Applied Optimization, University of Florida, Gainesville, FL, USA.

BILEVEL PARAMETRIC-EFFICIENT PORTFOLIO IN OUR EXAMPLE World Congress on Global Optimization, 2015, Centre of Applied Optimization, University of Florida, Gainesville, FL, USA.

STEP1. SUBSTITUTE THE FOLLOWER PROBLEM WITH ITS KKT OPTIMALITY CONDITIONS TO CONVERT THE BLP INTO A SINGLE LEVEL NLP World Congress on Global Optimization, 2015, Centre of Applied Optimization, University of Florida, Gainesville, FL, USA.

World Congress on Global Optimization, 2015, Centre of Applied Optimization, University of Florida, Gainesville, FL, USA.

World Congress on Global Optimization, 2015, Centre of Applied Optimization, University of Florida, Gainesville, FL, USA.

CONVERTED SINGLE LEVEL PARAMETRIC-EFFICIENT PORTFOLIO PROBLEM IN OUR EXAMPLE World Congress on Global Optimization, 2015, Centre of Applied Optimization, University of Florida, Gainesville, FL, USA.

RESULT OBTAINED BY SOLVING WITH LINGO 14 GLOBAL SOLVER ρ R V y1 y2 y3 c1 c2 c3 Cmax 1.2833 0.0181 1 0.1 0.14 0.12 0.10 0.2 0.11 0.08 0.3 0.07 0.4 0.05 0.5 1.2831 0.9994 0.0005 0.03 0.6 1.1167 0.0514 0.83 3.84 0.7 1.5 0.01 0.8 1.0833 0.0147 0.82 0.41 0.9 0.26 0.15 1.0 1.1163 0.0505 0.9899 0.86 0.09 2.0 0.95 3.0 1.1103 0.0119 0.8457 0.0224 0.02 4.0 1.2479 0.0073 0.8045 0.1102 0.0853 5.0 1.2469 0.0071 0.7993 0.1124 0.0883 World Congress on Global Optimization, 2015, Centre of Applied Optimization, University of Florida, Gainesville, FL, USA.

THE R vs V CURVE World Congress on Global Optimization, 2015, Centre of Applied Optimization, University of Florida, Gainesville, FL, USA.

PROBLEM OF NON-CONVEXITY OF THE SINGLE LEVEL PROBLEM The resulted single level NLP is non-convex due to involvement of bilinear terms in the complementary slackness conditions always and in the stationarity condition if g and h are nonlinear For bilevel programming where all the constraints of the follower are linear, the complementary slackness conditions are the only nonlinearities Bard and Moore [1990]; Edmunds and Bard [1991]; Vincente et al. [1994]; Visweswaran et al. [1996]; Shimizu et al. [1997] tackled linear quadratic BLPs with the following additional requirements so that follower level constraints form a convex polyhedron: F and G must be convex f must be quadratic g and h must be affine World Congress on Global Optimization, 2015, Centre of Applied Optimization, University of Florida, Gainesville, FL, USA.

PROBLEM OF NON-CONVEXITY OF THE SINGLE LEVEL PROBLEM World Congress on Global Optimization, 2015, Centre of Applied Optimization, University of Florida, Gainesville, FL, USA.

TOWARDS REMOVAL OF NON-CONVEXITY To tackle the general nonlinear BLP without convexity assumption of F, G, f and g without linearity assumption of H and h let us apply the technique of Gumus and Floudas [2001] which takes help of alphaBB algorithm of Adjiman et al. [1998a] for solving general twice-differentiable constrained NLPs World Congress on Global Optimization, 2015, Centre of Applied Optimization, University of Florida, Gainesville, FL, USA.

STEP 2. TO REMOVE THE NON-CONVEXITY DUE TO COMPLEMENTARY SLACKNESS CONDITIONS, EMPLOY THE ACTIVE SET STRATEGY FOR THE COMPLEMENTARY SLACKNESS CONDITIONS TO CONVERT THE SINGLE LEVEL NLP INTO A SINGLE LEVEL MIXED INTEGER NLP (MINLP) World Congress on Global Optimization, 2015, Centre of Applied Optimization, University of Florida, Gainesville, FL, USA.

World Congress on Global Optimization, 2015, Centre of Applied Optimization, University of Florida, Gainesville, FL, USA.

STEP 3. CONVERT THE MAXIMIZATION PROBLEMS TO MINIMIZATION PROBLEMS AND SOLVE THE MINLPS FOR THEIR LOCAL MINIMA WHICH WILL GIVE THE UPPER BOUNDS OF THE REQUIRED GLOBAL MINIMA World Congress on Global Optimization, 2015, Centre of Applied Optimization, University of Florida, Gainesville, FL, USA.

ρ R V y1 y2 y3 c1 c2 c3 Cmin 1.0833 0.0147 1 1.11 0.11 0.1 -0.1 1.13 0.2 1.15 0.3 0.33 0.12 0.4 1.2456 0.0118 0.8115 0.1885 0.09 0.06 0.5 1.2833 0.0181 0.07 0.04 0.6 1.2406 0.0027 0.7439 0.2561 0.05 1.53 0.7 0.8 0.03 0.9 0.02 1.0 2.0 1.1850 0.0074 0.4102 0.5898 0.24 3.0 4.0 5.0 World Congress on Global Optimization, 2015, Centre of Applied Optimization, University of Florida, Gainesville, FL, USA.

World Congress on Global Optimization, 2015, Centre of Applied Optimization, University of Florida, Gainesville, FL, USA.

STEP 4. CHECK IF THE LINEAR INDEPENDENCE CONDITION IS SATISFIED AT THE UPPER BOUND VALUES OBTAINED; IF SATISFIED GO TO THE NEXT STEP, ELSE STOP World Congress on Global Optimization, 2015, Centre of Applied Optimization, University of Florida, Gainesville, FL, USA.

World Congress on Global Optimization, 2015, Centre of Applied Optimization, University of Florida, Gainesville, FL, USA.

STEP 5. DECOMPOSE EACH NONLINEAR FUNCTION INTO A SUM OF LINEAR, BILINEAR, TRILINEAR, FRACTIONAL, FRACTIONAL TRILINEAR, CONVEX, UNIVARIATE CONCAVE, PRODUCT OF UNIVARIATE CONCAVES AND GENERAL NONCONVEX TERMS. THEN GENERATE CONVEX ENVELOPES FOR EACH OF BILINEAR, TRILINEAR, FRACTIONAL, FRACTIONAL TRILINEAR AND UNIVARIATE CONCAVE TERMS. World Congress on Global Optimization, 2015, Centre of Applied Optimization, University of Florida, Gainesville, FL, USA.

World Congress on Global Optimization, 2015, Centre of Applied Optimization, University of Florida, Gainesville, FL, USA.

World Congress on Global Optimization, 2015, Centre of Applied Optimization, University of Florida, Gainesville, FL, USA.

CONVEXIFIED PARAMETRIC-EFFICIENT PORTFOLIO PROBLEM World Congress on Global Optimization, 2015, Centre of Applied Optimization, University of Florida, Gainesville, FL, USA.

STEP 6. SOLVE THE CONVEXIFIED MINLPS TO GET THE LOWER BOUNDS OF THE REQUIRED GLOBAL MINIMA World Congress on Global Optimization, 2015, Centre of Applied Optimization, University of Florida, Gainesville, FL, USA.

ρ R V y1 y2 y3 c1 c2 c3 Cmin 1.1134 0.0427 0.9 0.1 -0.1 1.0833 0.0147 1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 1.0 2.0 3.0 4.0 5.0 World Congress on Global Optimization, 2015, Centre of Applied Optimization, University of Florida, Gainesville, FL, USA.

World Congress on Global Optimization, 2015, Centre of Applied Optimization, University of Florida, Gainesville, FL, USA.

STEP 7. CHECK THE UPPER BOUND AND THE LOWER BOUND OF THE OBJECTIVE FUNCTION; IF THEIR DIFFERENCE IS WITHIN A PERMISSIBLE RANGE, STOP; THE GLOBAL OPTIMUM IS OBTAINED; ELSE, EMPLOY BRANCHING ON A SELECTED VARIABLE THAT PARTICIPATES IN ONE OF THE NONLINEAR TERMS TO PARTITION THE INITIAL DOMAIN OF THE VARIABLE INTO TWO SUBDOMAINS IN ACCORDANCE WITH alphaBB GLOBAL OPTIMIZATION ALGORITHM World Congress on Global Optimization, 2015, Centre of Applied Optimization, University of Florida, Gainesville, FL, USA.

CONCLUSION In this problem of bilevel portfolio optimization we can see that both the upper bound and the lower bounds of the minimum of the objective function -C has come as - 0.1. Hence, we may conclude that the global minimum of -C is -0.1; i.e., the global maximum of C is 0.1. Hence, we stop here and conclude that the maximum total brokerage charge per dollar of investment should be 10 cents. World Congress on Global Optimization, 2015, Centre of Applied Optimization, University of Florida, Gainesville, FL, USA.

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