## Invited Session Fri.3.MA 004

#### Friday, 15:15 - 16:45 h, Room: MA 004

**Cluster 16: Nonlinear programming** [...]

### Fast gradient methods for nonlinear optimization and applications II

**Chair: William Hager**

**Friday, 15:15 - 15:40 h, Room: MA 004, Talk 1**

**Ya-Feng Liu**

Max-min fairness linear transceiver design for a multi-user MIMO interference channel

**Coauthors: Yu-Hong Dai, Zhi-Quan Tom Luo**

**Abstract:**

Consider the max-min fairness linear transceiver design problem for a multi-user multi-input multi-output (MIMO) interference channel. When the channel knowledge is perfectly known, this problem can be formulated as the maximization of the minimum signal to interference plus noise ratio (SINR) utility, subject to individual power constraints at each transmitter. We prove in this paper that, if the number of

antennas is at least two at each transmitter (receiver) and is at least three at each receiver (transmitter), the max-min fairness linear transceiver design problem is computationally intractable as the number of users becomes large. In fact, even the problem of checking the feasibility of a given set of target SINR levels is strongly NP-hard. We then propose two iterative algorithms to solve the max-min fairness linear transceiver design problem. The transceivers generated by these algorithms monotonically improve the min-rate utility

and are guaranteed to converge to a stationary solution. The efficiency and performance of the proposed algorithms

compare favorably with solutions obtained from the channel matched beamforming or the leakage interference minimization.

**Friday, 15:45 - 16:10 h, Room: MA 004, Talk 2**

**William Hager**

A primal-dual active set algorithm for nonlinear optimization with polyhedral constraints

**Coauthor: Hongchao Zhang**

**Abstract:**

A primal-dual active set algorithm is developed

for nonlinear optimization with polyhedral constraints.

The algorithm consists of a nonmonotone gradient projection

phase implemented by dual active set techniques,

an unconstrained optimization phase in the subspace

determined by the active set, and a set of rules for

branching between the two phases. Global convergence to

a stationary point is established. For a nondegenerate

stationary point, the algorithm eventually reduces to an

unconstrained optimization in a subspace without restarts.

Similarly, for a degenerate stationary point where

the strong second-order sufficient optimality

condition holds, the algorithm eventually reduces

to unconstrained optimization in a subspace. A specific

implementation of the algorithm is given which exploits

a new dual active set algorithm for the gradient projection

step and the conjugate gradient algorithm \texttt{CG\_{D}ESCENT} for

unconstrained optimization.

**Friday, 16:15 - 16:40 h, Room: MA 004, Talk 3**

**Yu-Hong Dai**

A perfect example for the BFGS method

**Abstract:**

Consider the BFGS quasi-Newton method applied to a general

non-convex function that has continuous second derivatives. This paper aims to construct a four-dimensional example such that the BFGS method need not converge. The example is perfect in the following sense: (a) All the stepsizes are exactly equal to one; the unit stepsize can also be accepted by various line searches including the Wolfe line search and the Arjimo line search; (b) The objective function is strongly convex along each search direction although it is not in itself. The unit stepsize is the unique minimizer of each line search function. Hence the example also applies to the global line search and the line search that always picks the first local minimizer; (c) The objective function is polynomial and hence is infinitely continuously differentiable. If relaxing the convexity requirement of the line search function; namely, (b), we are able to construct a relatively simple polynomial example.