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We show that a class of semidefinite programs (SDP) admits a solution that is a positive semidefinite
matrix of rank at most $r$, where $r$ is the rank of the matrix involved in the objective function of the SDP.
The optimization problems of this class are semidefinite packing problems,
which are the SDP analogs to vector packing problems.
Of particular interest is the case in which our result guarantees the existence of a solution
of rank one: we show that the computation of this solution actually reduces to a
Second Order Cone Program (SOCP).
We point out an application in statistics, in the optimal design of experiments.

In the past few years several applications of optimal
experimental designs have emerged to optimize the measurements
in communication networks. The optimal design problems arising from
this kind of applications share three interesting properties:
(i) measurements are only available at a small number of locations of the network;
(ii) each monitor can simultaneously measure several quantities, which
can be modeled by ``multiresponse experiments";
(iii) the observation matrices depend on the topology of the network.
In this paper, we give an overview of these experimental design
problems and recall recent results for the computation of optimal
designs by Second Order Cone Programming (SOCP). New results for the
network-monitoring of a discrete time process are presented. In particular, we show
that the optimal design problem for the monitoring of an AR1 process can be reduced
to the standard form and we give experimental results.

We study a family of combinatorial optimization problems
defined by a parameter $p\in[0,1]$, which involves spectral
functions applied to positive semidefinite matrices, and has
some application in the theory of optimal experimental design.
This family of problems tends to a generalization of the classical
maximum coverage problem as $p$ goes to $0$, and to a trivial instance
of the knapsack problem as $p$ goes to $1$.
In this article, we establish a matrix inequality which shows that the objective function is submodular for all $p\in[0,1]$, from which it follows
that the greedy approach, which has often been used for this problem, always gives a design within $1-1/e$ of the optimum.
We next study the design found by rounding the solution of the continuous relaxed problem, an approach which has been applied by several authors.
We prove an inequality which generalizes a classical result from the theory
of optimal designs, and allows us to give a rounding procedure with an approximation
factor which tends to $1$ as $p$ goes to $1$.

PICOS is a user friendly interface
to several conic and integer programming solvers,
very much like YALMIP
under MATLAB.
The main motivation for PICOS is to have the possibility to
enter an optimization problem as a high level model,
and to be able to solve it with several different solvers.
Multidimensional and matrix variables are handled in a natural fashion,
which makes it painless to formulate a SDP or a SOCP.
This is very useful for educational purposes,
and to quickly implement some models and
test their validity on simple examples.
Furthermore, with PICOS you can take advantage of the
python programming language to read and write data,
construct a list of constraints by using python list comprehensions,
take slices of multidimensional variables, etc.

We present a new semidefinite representation for the trace of
a real function f applied to symmetric matrices, when a
semidefinite representation of the convex function f is known. Our construction
is intuitive, and yields a representation that is more compact than the previously known one.
We also show with the help of matrix geometric means and the Riemannian metric of the set of positive definite matrices
that for a rational number p in the interval (0,1],
the matrix X raised to the exponent p is the largest element
of a set represented by linear matrix inequalities.
We give numerical results for a problem inspired from the theory
of experimental designs, which show that the new semidefinite programming formulation
yields a speed-up factor in the order of 10.

Let $G$ be a directed acyclic graph with $n$ arcs, a source $s$ and a sink $t$. We introduce the cone $K$ of flow matrices, which is a polyhedral cone
generated by the matrices $1_P 1_P^T \in R^{n\times n}$, where
$1_P\in R^n$ is the incidence vector of the $(s,t)$-path $P$.
Several combinatorial problems reduce to a linear optimization problem over $K$.
This cone is intractable, but we provide two convergent approximation hierarchies, one of them based on a
completely positive representation of $K$.
We illustrate this approach by computing bounds for a maximum flow problem with pairwise arc-capacities.

An algorithm based on a delayed constraint generation method for solving semi-infinite programs
for constructing minimax optimal designs for nonlinear models is proposed. The outer optimization level of the minimax
optimization problem is solved using a semidefinite programming based approach that requires
the design space be discretized. A nonlinear programming solver is then used to solve the inner program
to determine the combination of the parameters that yields the worst-case value of the design criterion.
The proposed algorithm is applied to find minimax optimal designs for the logistic model, the flexible 4-parameter
Hill homoscedastic model and the general nth order consecutive reaction model, and shows that it
(i) produces designs that compare well with minimax $D-$optimal designs obtained from semi-infinite programming method in the literature;
(ii) can be applied to semidefinite representable optimality criteria, that include the common A-, E-,G-, I- and D-optimality criteria;
(iii) can tackle design problems with arbitrary linear constraints on the weights; and
(iv) is fast and relatively easy to use.

We propose (Mixed Integer) Second Order Cone Programming formulations to find approximate and exact $D-$optimal designs for $2^k$
factorial experiments for Generalized Linear Models (GLMs). Locally optimal designs are addressed with Second Order Cone Programming
(SOCP) and Mixed Integer Second Order Cone Programming (MISOCP) formulations.
The formulations are extended for scenarios of parametric uncertainty employing the Bayesian framework for
\emph{log det} $D-$optimality criterion. A quasi Monte-Carlo sampling procedure based
on the Hammersley sequence is used for integrating the optimality criterion in the parametric region. The problems are solved in \texttt{GAMS}
environment using \texttt{CPLEX} solver. We demonstrate the application of the algorithm with the logistic, probit and complementary log-log models
and consider full and fractional factorial designs.

Let G be a directed acyclic graph with n arcs, a source s and a sink t. We introduce the cone K of flow matrices, which is a polyhedral cone
generated by the matrices $\vec{1}_P\vec{1}_P^T\in\RR^{n\times n}$, where
$\vec{1}_P\in\RR^n$ is the incidence vector of the (s,t)-path P.
We show that several hard flow (or path) optimization problems, that cannot be solved by using the standard arc-representation
of a flow, reduce to a linear optimization problem over $\mathcal{K}$.
This cone is intractable: we prove that the membership problem associated to $\mathcal{K}$
is NP-complete. However, the affine hull of this cone admits a nice description,
and we give an algorithm which computes in polynomial-time the decomposition of a matrix
$X\in \operatorname{span} \mathcal{K}$ as a linear combination of some $\vec{1}_P\vec{1}_P^T$'s.
Then, we provide two convergent approximation hierarchies, one of them based on a
completely positive representation of~K.
We illustrate this approach by computing bounds for
the quadratic shortest path problem, as well as
a maximum flow problem with pairwise arc-capacities.