Approximation of the Whole Pareto-Optimal Set for the Vector Optimization Problem

In multi objective optimization problems several objective functions have to be minimized simultaneously. In this work, we present a new computational method for the numerical solution of the linearly constrained, convex multi objective optimization problem. We propose some technique to find joint decreasing direction for unconstrained and linearly constrained case as well. Based on these results we introduce a method using subdivision technique to approximate the whole Pareto-optimal set of the linearly constrained, convex multi objective optimization problem. Finally, we illustrate computations of our algorithm by solving the Markowitz-model on real data

Predictor-corrector interior-point algorithm for sufficient linear complementarity problems based on a new type of algebraic equivalent transformation technique

We propose a new predictor-corrector (PC) interior-point algorithm (IPA) for solving linear complementarity problem (LCP) with P_* (κ)-matrices. The introduced IPA uses a new type of algebraic equivalent transformation (AET) on the centering equations of the system defining the central path. The new technique was introduced by Darvay et al. [21] for linear optimization. The search direction discussed in this paper can be derived from positive-asymptotic kernel function using the function φ(t)=t^2 in the new type of AET. We prove that the IPA has O(1+4κ)√n log⁡〖(3nμ^0)/ε〗 iteration complexity, where κ is an upper bound of the handicap of the input matrix. To the best of our knowledge, this is the first PC IPA for P_* (κ)-LCPs which is based on this search direction

Approximate dynamic programming for large scale systems

Sequential decision making under uncertainty is at the heart of a wide variety of practical problems. These problems can be cast as dynamic programs and the optimal value function can be computed by solving Bellman's equation. However, this approach is limited in its applicability. As the number of state variables increases, the state space size grows exponentially, a phenomenon known as the curse of dimensionality, rendering the standard dynamic programming approach impractical. An effective way of addressing curse of dimensionality is through parameterized value function approximation. Such an approximation is determined by relatively small number of parameters and serves as an estimate of the optimal value function. But in order for this approach to be effective, we need Approximate Dynamic Programming (ADP) algorithms that can deliver `good' approximation to the optimal value function and such an approximation can then be used to derive policies for effective decision-making. From a practical standpoint, in order to assess the effectiveness of such an approximation, there is also a need for methods that give a sense for the suboptimality of a policy. This thesis is an attempt to address both these issues. First, we introduce a new ADP algorithm based on linear programming, to compute value function approximations. LP approaches to approximate DP have typically relied on a natural `projection' of a well studied linear program for exact dynamic programming. Such programs restrict attention to approximations that are lower bounds to the optimal cost-to-go function. Our program -- the `smoothed approximate linear program' -- is distinct from such approaches and relaxes the restriction to lower bounding approximations in an appropriate fashion while remaining computationally tractable. The resulting program enjoys strong approximation guarantees and is shown to perform well in numerical experiments with the game of Tetris and queueing network control problem. Next, we consider optimal stopping problems with applications to pricing of high-dimensional American options. We introduce the pathwise optimization (PO) method: a new convex optimization procedure to produce upper and lower bounds on the optimal value (the `price') of high-dimensional optimal stopping problems. The PO method builds on a dual characterization of optimal stopping problems as optimization problems over the space of martingales, which we dub the martingale duality approach. We demonstrate via numerical experiments that the PO method produces upper bounds and lower bounds (via suboptimal exercise policies) of a quality comparable with state-of-the-art approaches. Further, we develop an approximation theory relevant to martingale duality approaches in general and the PO method in particular. Finally, we consider a broad class of MDPs and introduce a new tractable method for computing bounds by consider information relaxation and introducing penalty. The method delivers tight bounds by identifying the best penalty function among a parameterized class of penalty functions. We implement our method on a high-dimensional financial application, namely, optimal execution and demonstrate the practical value of the method vis-a-vis competing methods available in the literature. In addition, we provide theory to show that bounds generated by our method are provably tighter than some of the other available approaches

An Algorithm for Biobjective Mixed Integer Quadratic Programs

Multiobjective quadratic programs (MOQPs) are appealing since convex quadratic programs have elegant mathematical properties and model important applications. Adding mixed-integer variables extends their applicability while the resulting programs become global optimization problems. Thus, in this work, we develop a branch and bound (BB) algorithm for solving biobjective mixed-integer quadratic programs (BOMIQPs). An algorithm of this type does not exist in the literature. The algorithm relies on five fundamental components of the BB scheme: calculating an initial set of efficient solutions with associated Pareto points, solving node problems, fathoming, branching, and set dominance. Considering the properties of the Pareto set of BOMIQPs, two new fathoming rules are proposed. An extended branching module is suggested to cooperate with the node problem solver. A procedure to make the dominance decision between two Pareto sets with limited information is proposed. This set dominance procedure can eliminate the dominated points and eventually produce the Pareto set of the BOMIQP. Numerical examples are provided. Solving multiobjective quadratic programs (MOQPs) is fundamental to our research. Therefore, we examine the algorithms for this class of problems with different perspectives. The scalarization techniques for (strictly) convex MOPs are reviewed and the available algorithms for computing efficient solutions for MOQPs are discussed. These algorithms are compared with respect to four properties of MOQPs. In addition, methods for solving parametric multiobjective quadratic programs are studied. Computational studies are provided with synthetic instances, and examples in statistics and portfolio optimization. The real-life context reveals the interplay between the scalarizations and provides an additional insight into the obtained parametric solution sets

Allocation of Classroom Space Using Linear Programming (A Case Study: Premier Nurses Training College, Kumasi)

The use of linear programming to solve the problem of over-allocation and under-allocation of the scarce classroom space was considered with particular reference to the Premier Nurse’s Training College, Kumasi. Data was collected from the College on the classroom facilities and the number of students per programme. A linear programming model was formulated based on the data collected to maximize the usage of the limited classroom space. POM-QM for Windows 4 (Software for Quantitative Methods, Production and Operation Management by Howard J. Weiss) was used based on the simplex algorithm to obtain optimal solution.Analysis of the results showed that six (50%) of the twelve classrooms could be used to create a maximum classroom space of six hundred and forty.  It was also observed that the management could use two hundred and eighty (280) surplus spaces to increase its student’s intake from three hundred and sixty (360) to six hundred and forty (640) students, an increase of about 77.78% with only 50% of the total number of classrooms. Again management could cut down the number of classrooms used from twelve to six and reduce the cost of maintaining the classrooms by 50% and still have as many as six extra classrooms for other equally important purposes, hence maximize its profit margin. Keywords: Linear programming, Allocation, Optimal solution, Simplex Algorithm, Premier Nurse’s Training College.

Experiments with Active-Set LP Algorithms Allowing Basis Deficiency

n interesting question for linear programming (LP) algorithms is how to deal with solutions in which the number of nonzero variables is less than the number of rows of the matrix in standard form. An approach is that of basis deficiency-allowing (BDA) simplex variations, which work with a subset of independent columns of the coefficient matrix in standard form, wherein the basis is not necessarily represented by a square matrix. We describe one such algorithm with several variants. The research question deals with studying the computational behaviour by using small, extreme cases. For these instances, we must wonder which parameter setting or variants are more appropriate. We compare the setting of two nonsimplex active-set methods with Holmström’s TomLab LpSimplex v3.0 commercial sparse primal simplex commercial implementation. All of them update a sparse QR factorization in Matlab. The first two implementations require fewer iterations and provide better solution quality and running time.This work has been funded by grant PID2021-123278OB-I00 from the Spanish Ministry of Science and Innovation. Partial funding for open access charge: Universidad de Málag

International Conference on Continuous Optimization (ICCOPT) 2019 Conference Book

The Sixth International Conference on Continuous Optimization took place on the campus of the Technical University of Berlin, August 3-8, 2019. The ICCOPT is a flagship conference of the Mathematical Optimization Society (MOS), organized every three years. ICCOPT 2019 was hosted by the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin. It included a Summer School and a Conference with a series of plenary and semi-plenary talks, organized and contributed sessions, and poster sessions. This book comprises the full conference program. It contains, in particular, the scientific program in survey style as well as with all details, and information on the social program, the venue, special meetings, and more

New predictor-corrector interior-point algorithm with AET function having inflection point

In this paper we introduce a new predictor-corrector interior-point algorithm for solving P_* (κ)-linear complementarity problems. For the determination of search directions we use the algebraically equivalent transformation (AET) technique. In this method we apply the function φ(t)=t^2-t+√t which has inflection point. It is interesting that the kernel corresponding to this AET function is neither self-regular, nor eligible. We present the complexity analysis of the proposed interior-point algorithm and we show that it's iteration bound matches the best known iteration bound for this type of PC IPAs given in the literature. It should be mentioned that usually the iteration bound is given for a fixed update and proximity parameter. In this paper we provide a set of parameters for which the PC IPA is well defined. Moreover, we also show the efficiency of the algorithm by providing numerical results