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**1 - 4**of**4**### QUADRATIC PROGRAMMING (QP) ALGORITHMS & MEAN-VARIANCE PORTFOLIO OPTIMISATION

, 2002

"... QP is the optimization of a quadratic function subject to linear equality and inequality constraints. It arises in multiple objective decision making where the departure of the actual decisions from their corresponding ideal, or bliss, value can be evaluated using a weighted quadratic norm as a meas ..."

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QP is the optimization of a quadratic function subject to linear equality and inequality constraints. It arises in multiple objective decision making where the departure of the actual decisions from their corresponding ideal, or bliss, value can be evaluated using a weighted quadratic norm as a measure of deviation. The formulation of mean-variance optimization of uncertain systems also leads to QP. An important application of mean-variance is in simple optimal portfolio problems where the constraints are linear and the objective function is quadratic (Markowitz, 1959). The decision maker has to reconcile the conflicting desires of maximizing expected portfolio return, represented by the linear term, and minimizing the portfolio risk, represented by the quadratic (variance) term, in the objective function. Sequential QP algorithms require the solution of QP subproblems to generate descent directions for general nonlinear optimization and minimax.

### SEMIDEFINITE PROGRAMMING*

"... Abstract. In sernidefinite programming, one minimizes a linear function subject to the constraint that an affine combination of synunetric matrices is positive semidefinite. Such a constraint is nonlinear and nonsmooth, but convex, so semidefinite programs are convex optimization problems. Semidefin ..."

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Abstract. In sernidefinite programming, one minimizes a linear function subject to the constraint that an affine combination of synunetric matrices is positive semidefinite. Such a constraint is nonlinear and nonsmooth, but convex, so semidefinite programs are convex optimization problems. Semidefinite programming unifies several standard problems (e.g., linear and quadratic programming) and finds many applications in engineering and combinatorial optimization. Although semidefinite programs are much more general than linear programs, they are not much harder to solve. Most interior-point methods for linear programming have been generalized to semidefinite programs. As in linear programming, these methods have polynomial worst-case complexity and perform very well in practice. This paper gives a survey of the theory and applications of semidefinite programs and an introduction to primaldual interior-point methods for their solution. Key words, semidefinite programming, convex optimization, interior-point methods, eigenvalue optimization, combinatorial optimization, system and control theory AMS subject classifications. 65K05, 49M45, 93B51, 90C25, 90C27, 90C90, 15A18 1. Introduction. 1.1. Semidefinite programming. We consider the problem of minimizing a linear function