Decision variables can be continuous represented by real numbers , resulting in continuous optimization problems, or discrete represented by integer numbers , resulting in integer optimization also called combinatorial optimization problems. In many instances, there is a mix of continuous and integer decision variables.
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As an illustrative example, consider the "diet problem", one of the first modern optimization problems , studied in the s: to find the cheapest combination of foods that will satisfy all the daily nutritional requirements of a person. In this classical problem, the objective function to minimize is the cost of the food, the decision variables are the amounts of each type of food to be purchased assumed as continuous variables , and the constraints are the nutritional needs be satisfied, like total calories, or amounts of vitamins, minerals, etc. The "diet problem" has certain interesting properties: it is a continuous problem where both the objective function total cost, i.
These linear constraints define a feasible space space of decision variables where constraints are satisfied which is a convex polyhedron, so it is a convex problem. Convex optimization problems  are particularly interesting, since they have a unique solution i. Non linear programming NLP deals with continuous problems where some of the constraints or the objective function are nonlinear. Further, the presence of nonlinearities in the objective and constraints might imply nonconvexity, which results in the potential existence of multiple local solutions multimodality.
Thus, in nonconvex problems one should seek the globally optimal solution among the set of possible local solutions. For the simple case of only two decision variables, one can visualize the objective function of a multimodal problem as a terrain with multiple peaks.
Simple examples of unimodal and multimodal surfaces are presented in Figure 1. The solution of multimodal problems is studied by the subfield of global optimization . Many continuous problems and the vast majority of combinatorial optimization problems belong to this class. Most problems in global optimization are very hard to solve exactly in a reasonable computation time.
Fortunately, recent developments indicate that convex optimization problems are more prevalent in practice than was previously thought . Thus, it is highly desirable to formulate or reformulate the statement of any optimization problem as a convex one. The book by Boyd and Vandenberghe  gives detailed information on how to recognize, formulate, and solve convex optimization problems. Model-based optimization is a key methodology in engineering, helping in the design, analysis, construction and operation of all kind of devices.
Since engineering approaches are playing a significant role in the rapid evolution of systems biology , it is expected that mathematical optimization methods will contribute in a significant way to advances in systems biology. Examples of applications of optimization in systems biology, classified by the type of optimization problem, are given.
Below, I highlight several topics where optimization has already made significant contributions. Optimization methods have been applied in both metabolic control analysis [15,16] and biochemical systems theory . Further, optimization and, more in particular, linear programming has been the engine behind metabolic flux balance analysis, where the optimal flux distributions are calculated using linear optimization, and are used to represent the metabolic phenotype for certain conditions.
This flux balance methodology provides a guide to metabolic engineering and a method for bioproc-ess optimization . Examples of success stories are the in silico predictions of Escherichia coli metabolic capabilities , or the genome-scale reconstruction of the Sac-charomyces cerevisiae metabolic network .
Metabolic engineering exploits an integrated, systemslevel approach for optimizing a desired cellular property or phenotype . New optimization-based methods are being developed by using genome-scale metabolic models, which enable identification of gene knockout strategies for obtaining improved phenotypes. However, these problems have a combinatorial nature, so the computational time increases exponentially with the size of the problem for exact methods, so there is a clear need of developing approximate yet faster algorithms .
Not surprisingly, optimization will also help in the bioengineering of novel in vitro metabolic pathways using synthetic biology, as the key component in rational redesign and directed evolution . Coupling constraint-based analysis with optimization has been used to generate a consistent framework for the generation of hypotheses and the testing of functions of microbial cells using genome-scale models .
Extensions and modifications of flux balance analysis continue to use optimization methods extensively . A particularly interesting question in this context concerns the principles behind the optimal metabolic network operation, i. Constrained evolutionary optimization has also been used to understand optimal circuit design . Moreover, optimization principles have also been used to explain the complexity and robustness found in biochemical networks , and much more work in this topic is to be expected in the near future.
Related to this, the hypothesis that metabolic systems have evolved optimal strategies as a result of evolutionary pressures has been used in cybernetic models , an approach which may offer advantages over traditional methodologies. Simple examples two decision variables, no constraints of unimodal l. Table 1: Examples of applications of optimization in systems biology, classified by type of optimization problem note that several types overlap.
Optimization with differential equations as constraints and possible time-dependent decision variables. Reverse engineering in systems biology aims to reconstruct the biochemical interactions from data sets of a particular biological system. Optimization has been used for inferring important biomolecular networks, such as e. System identification [50,51] is a methodology widely used in engineering for building mathematical models of dynamical systems based on measured data.
Roughly, this involves selected the structure of the model and estimating the parameters of such model from the available experimental data. The problem of parameter estimation in biochemical pathways, formulated as a nonlinear programming problem subject to the pathway model acting as constraints, has also received great attention . Since these problems are frequently multimodal, global optimization methods are needed in order to avoid local solutions.
Handbook of Applied Optimization
A local solution can be very misleading when calibrating models: it would indicate a bad fit even for a model which could potentially match perfectly a set of experimental data. Since biological experiments are both expensive and time consuming, it would be ideal if one could plan them in an optimal way, i. This is the purpose of optimal experimental. Although, as already mentioned, it would be desirable to formulate all the optimization problems as convex ones, in many occasions this is not possible, so we face the solution of global optimization problems, most of which belong to the class of NP-hard problems , where obtaining global optima with guarantees will be impossible in many instances.
In these situations, approximate techniques like stochastic global optimization can at least locate a near globally optimal solution in a reasonable time, although the cost to pay is that these methods do not offer full guarantees of global optimality. In this context, evolutionary computation methods are a class of stochastic methods which have shown good performance in systems biology applications [55,]. Hybrid methods, combining global and local techniques, have also shown great potential with difficult problems like parameter estimation [54,59,70].
Much more work is needed to further enhance the efficiency and robustness of these approaches in order to make then applicable to large scale models. Another important issue is the stochasticity that is inherent in biomolecular systems [71,72]. This stochastic nature requires advances in optimization methods, and a number of researches are already providing useful approaches, such as in parameter estimation in stochastic biochemical reactions  or in the optimization of stochastic gene network models .
As stated in , it would be desirable to have computer-aided design tools for biological engineering, similarly to what already happens in many other areas of engineering. Such software would guide the improvement of the behaviour of a biological system in silico by optimizing design parameters targeting a selected objective function. The optimization of such synthetic biological systems is in fact receiving increasing attention: optimization algorithms could search for the components promoters, operators, regulatory proteins, inducers, etc.
A promising example of what can be done is the OptCircuit framework , which can be used as an optimization-based design platform to aid in the construction and fine tuning of integrated biological circuits. Other researches are adapting the workflow developed by the electronics industry to the design and assembly of very large scale integrated genetic systems, claiming that the computer. Moreover, optimization could also be used after the design and construction phases, inside a model predictive control framework , to optimally manipulate the resulting biological systems.
This is the dream of metabolic engineering [26,79] and synthetic biology [21,25,74]. We are still not there, but the purpose of this paper has been to show that we are getting close. Several issues must be addressed before we reach that goal. First, we need robust and efficient methods for optimization under uncertainty, and for the optimization of stochastic models, that are also able to scale-up, hopefully even at the level of genome-scale models. Second, since neither we nor nature rarely have a single objective, we need mul-ticriteria optimization methods that are better able to cope with the scale and complexity of models from systems biology .
Finally, it should be recognized that standard optimization can be sometimes insufficient for gaining deeper insights regarding certain aspects of systems biology, such as in the evolution of biological systems. While evolving towards optimal properties, the environment may change or organisms may even change their own environment, which in turn alters the optimum.
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American Institute of Mathematical Sciences. Previous Article Solvability of a class of thermal dynamical contact problems with subdifferential conditions. Global optimization via differential evolution with automatic termination. In this survey, univariate global optimization problems are considered where the objective function or its first derivative can be multiextremal black-box costly functions satisfying the Lipschitz condition over an interval. Such problems are frequently encountered in practice.
A number of geometric methods based on constructing auxiliary functions with the usage of different estimates of the Lipschitz constants are described in the paper. Keywords: geometric approach. Citation: Dmitri E. Kvasov, Yaroslav D. Univariate geometric Lipschitz global optimization algorithms.
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