**Wednesday, 16:15 - 16:40 h, Room: MA 041**

**Hyemin Jeon**

Convex quadratic programming with variable bounds

**Coauthors: Jeffrey T. Linderoth, Andrew J. Miller**

**Abstract:**

The set *X= {(x,z,v) ∈ ***R**_{+}^{n} × **B**^{n} × **R**_{+} | v ≥ x^{T}Qx, x_{j} ≤ z_{j} ∀ j } for some matrix *Q \succeq 0*

appears as substructure in many applications including portfolio management and data mining.

We aim to obtain a good approximation of *\operatorname{conv}(X)*, and our approach starts by reformulating the set using Cholesky factorization *Q = LL*^{T}.

In the reformulated set * S = {(y,t,z,v) ∈ ***R**^{n} × **R**_{+}^{n} × **B**^{n} × **R**_{+} |

v ≥ ∑_{j}t_{j}, t_{j} ≥ y_{j}^{2} ∀ j, 0 ≤ [L^{-T}y]_{j} ≤ z_{j} ∀ j },

the nonlinear constraints are convex and separable but the interaction between continuous and binary variables is more complicated.

Our work thus far has focused on studying the set *S* in the case *n=2* denoted by *S*_{2}.

A number of valid inequalities for *S*_{2} are derived, most of which are represented as second-order cone constraints.

Computational experiments are conducted to empirically compare the obtained relaxation to *\operatorname{conv}(S*_{2}),

and to demonstrate how to utilize our valid inequalities for the case *n>2*.

Talk 3 of the invited session Wed.3.MA 041

**"Quadratic integer programming"** [...]

Cluster 14

**"Mixed-integer nonlinear programming"** [...]