Tutorial Program Recent Developments of Optimization in Process Systems Engineering

Wednesday 17 January

09.30–10.10 Registration with coffee & sandwich

10.10–10.15 Opening words

10.15–11.55 Modeling in Process Control and Systems Engineering from a Sparse Perspective Hannu Toivonen and Mikael Manngård, Åbo Akademi

12.00–13.00 Lunch

13.10–14.10 Optimization in Process Systems Engineering Jan Kronqvist, Åbo Akademi

14.15–15.15 Semidefinite Programming — Basics Ray Pörn, Åbo Akademi

15.15–15.30 Coffee break

15.30–17.00 Semidefinite Programming — Advanced Topics and Applications Ray Pörn, Åbo Akademi

19.00– Dinner

You are encouraged to bring your own laptop to the tutorial with programs and toolboxes installed as explained in the topic outlines.

NPCW Tutorial, 17 January 2018, Åbo, Finland 1

Sparse modeling in process control and process systems engineering

Hannu Toivonen and Mikael Manngård

Developments in sparse optimization during the last twenty years have made it possible to address

certain combinatorial optimization problems, which were previously deemed intractable. The

purpose of this tutorial is to describe some applications of sparse optimization in process control and

systems engineering. These include variable selection for predictive models, identification of

switching systems, system identification in the presence of trends, and identification of low-order

dynamic models.

The structure of the presentation is as follows:

1) Background: examples of combinatorial optimization problems

- Variable selection, i.e., finding a small subset from a given set of variables which explains

data. For example, for predictive models, or to explain fault situations.

- Change detection in data or model. For example: identification of switching systems, which

switch between a number of modes at unknown time instants, or estimation of piecewise

linear trends in data.

2) A basic problem: selection of independent variables in regression model

Here we describe how a combinatorial problem can be solved, either exactly or approximately,

by relaxing it to a convex l1-constrained problem. The most important properties of this

approach are reviewed.

3) System identification in the presence of trends and level shifts

Here we identify a system model and structured disturbances (such as trends, outliers and level

shifts) simultaneously using sparse optimization. Results are demonstrated on numerical

examples.

4) Identification of low-order system models

System order can be characterized in terms of the rank of a Hankel matrix associated with the

system’s impulse response. Identification of a low-order model can therefore be formulated as a

rank-constrained optimization problem. These are numerically hard problems, but can be

relaxed by replacing the matrix rank by its nuclear norm, defined as the sum of the matrix

singular values. The nuclear norm relaxation results in a convex optimization problem, for which

efficient solvers exist.

For the numerical examples in the tutorial the cvx toolbox in Matlab will be used.

NPCW Tutorial, 17 January 2018, Åbo, Finland 2

Optimization in process systems engineering

Jan Kronqvist

In this workshop we present some important types of optimization problems, with focus on how to solve these problems. We present some basic theory, methods and some problem formulations. We demonstrate how these problems can be solved efficiently in Matlab by the state-of-the-art solvers Gurobi and IPOPT.

1) A brief introduction to optimization

2) Integer programmingWe present techniques for solving linear problems containing integer variables, mainly thebranch and bound method and some cutting planes.

3) Disjunctive programmingHow to incorporate logic decisions in optimizations problems.

4) Solving optimization problems in MatlabWe show how to solve linear problems with integer variables in Matlab using Gurobi. We alsogive a brief illustration on how to use the nonlinear solver IPOPT with Matlab.

5) Optimization problems with nonlinear functionsHow can we solve optimization problems with both nonlinear functions and integer variables?

In this workshop we will use the solvers Gurobi and IPOPT in Matlab. There are free Academic licenses available for both solvers and we encourage participant to obtain these in advance.

Opti toolbox which contains IPOPT and some other powerful solvers can be obtained from: https://www.inverseproblem.co.nz/OPTI/

Gurobi which is one of the most powerful solvers for linear and quadratic optimization problems can be obtained directly from: http://www.gurobi.com/ There is an interface to Matlab which is easy to set up. Obtaining the Academic license takes less than 5 minutes (can only be done with a university network connection).

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Semidefinite programming – basics and applications

Ray Pörn

Semidefinite programming (SDP) is one of the most exciting developments in mathematical

programming during the last 20 years. SDP has applications in very diverse fields such as convex

constrained optimization, control theory, combinatorial optimization, statistics, differential geometry

and many more. SDPs are often solved using interior-point methods and many applications can be

solved fairly efficiently in practice. The structure of this workshop is as follows:

1) Introduction and some applications

2) Representability

What can be expressed as SDPs? Many different things can be expressed using linear matrix

inequalities (LMIs) and solved with SDP methods. Some tools from linear algebra are given in

order to represent different constraints in correct form.

3) Relaxation and randomization

SDP provides tight approximation of different non-convex optimization problems. The SDP

solution can also be used as a basis for a fast randomization procedure in order to obtain a good

quality solution for the non-convex problem.

4) Reformulation

Combinatorial problems can be reformulated in many ways, a typical technique is linearization.

SDP can be used to obtain an optimal reformulation of certain combinatorial problem.

5) More applications

6) Computer exercises

The participants are encouraged to bring their own laptop with matlab and the cvx toolbox

(http://cvxr.com/cvx/download/) installed.

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