The "traditional" portfolio construction problem is...max U(w,L1) = r_p + L1*v_p...with U = goal (or utility) function, r_p = w'*r_a = portfolio return, w = vector of portfolio constituent weights, r_a = portfolio asset returns, v_p = portfolio risk (in the most basic case, this is portfolio variance) and L1 goal weight (or portfolio risk aversion coefficient, if you prefer). Plus constraints (linear and equality and inequality constraints, in the most basic case)This problem can be called a "bi-criteria" problem, because the goal function considers two criteria: risk and return. In practical investment management, additional criteria might be relevant (for example, asset liquidity or sustainability characteristics). If we introduce a third criteria c_p which is linear in constituent weights in the sense of c_p = w'*c_a, the problem becomes...max U'(w,L1,L2) = r_p + L1*v_p + L2*c_pMy question: Is the following logic correct"If we know L2 = L2' in advance, then it follows that...max U''(w,L1,L2') = w'*r_a + L2*v_p + L2'*w'*c_a = w'*(r_a + L2'*c_a) + L2*v_p = w'*r_a' + L2*v_pProblem U'' is a quadratic programming exercise if v_p is the portfolio variance. The only difference to the bi-criteria problem is that portfolio constituent returns r_a are replaced with r_a', which on constituent level i are calculated as "excess premia" r_ai' = r_ai - L2*c_ai if the third asset characteristics is a "bad" like in the case of an illiquidity measure."Discussions of portfolio problems with a second quadratic criteria can be found in the literature, but not tri-criteria problems with a second linear criteria, as far as I am aware of.Greetings,Andi

yes

Thank you for confirming this.It seems to be the case that the resulting tri-criteria frontier plotted in mean-variance space is not concave throughout anymore. The consequence is that certain corner solutions will dominate for a wide range of (volatility-) risk aversions. See example below which features sustainability scores and sustainability preference (expressed as a % weight between the financial expected returns and sustainability characteristics). Of course, the resulting frontier will be well-behaved and concave in "sustainability-adjusted-mean-variance space".GreetingsAndi