MATHEMATICS OF OPERATIONS RESEARCH Vol. 00, No. 0, Xxxxx 0000, pp. 000–000 issn 0364-765X | eissn 1526-5471 | 00 | 0000 | 0001 INFORMS doi 10.1287/xxxx.0000.0000 c  0000 INFORMS Authors are encouraged to submit new papers to INFORMS journals by means of a style file template, which includes the journal title. However, use of a template does not certify that the paper has been accepted for publication in the named jour- nal. INFORMS journal templates are for the exclusive purpose of submitting to an INFORMS journal and should not be used to distribute the papers in print or online or to submit the papers to another publication. K-optimal design via semidefinite programming and entropy optimization Pierre Maréchal Institut de Mathématiques de Toulouse, Université de Toulouse, France pr.marechal@gmail.com, http://www.math.univ-toulouse.fr/ marechal/ Jane J. Ye and Julie Zhou Department of Mathematics and Statistics, University of Victoria, Victoria, B.C. Canada V8W 2Y2, janeye@uvic.ca,jzhou@uvic.ca, http://www.math.uvic.ca/faculty/janeye/,http://www.math.uvic.ca/ jzhou/ In this paper, we consider the problem of optimal design of experiments. A two step inference strategy is proposed. The first step consists in minimizing the condition number of the so-called information matrix. This step can be turned into a semidefinite programming (SDP) problem. The second step is more classical, and it entails the minimization of a convex integral functional under linear constraints. This step is formulated in some infinite-dimensional space and is solved by means of a dual approach. Numerical simulations will show the relevance of our approach. Key words : Optimal design of experiments, condition numbers, semidefinite programming, entropy optimization, Fenchel duality, Chebyshev polynomials. MSC2000 subject classification : Primary: 90C26, 90C30, 62K05; secondary: 90C22 OR/MS subject classification : Primary:design of experiments, entropy; secondary:nondifferentiable, programming, algorithms History : Date submitted:7/29/2012;Dates revised: 7/23/2013,3/23/2014 1. Introduction. We consider the parametric regression model y = p  j=0 θ j f j (x)+ ε, x ∈ S, in which y ∈ R is the response variable, x ∈ S ⊂ R k is the design variable, ε is a random error with mean 0 and variance σ 2 , f j , j =0,...,p are given basis functions supported in S and the θ j , j =0,...,p are parameters to be estimated. Example 1.1. If S =[−1, 1] and f j (x)= x j as in [28], one speaks of p-th order polynomial regression on [−1, 1]. We shall use the notation f (x)=(f 0 (x),...,f p (x)) ⊤ and θ =(θ 0 ,...,θ p ) ⊤ . 1