Baha Alzalg

• Professor
• Dept of Mathematics at University of Jordan

Jin et al. (in J. Optim. Theory Appl. 155:1073-1083, 2012) proposed homogeneous self-dual algorithms for stochastic semidefinite programs with finite event space. In this paper, we utilize their work to derive homogeneous self-dual algorithms for stochastic second-order cone programs with finite event space. We also show how the structure in the st...

We introduce and study two-stage stochastic symmetric programs with recourse to handle uncertainty in data defining (deterministic) symmetric programs in which a linear function is minimized over the intersection of an affine set and a symmetric cone. We present a Benders' decomposition-based interior point algorithm for solving these problems and...

Ariyawansa and Zhu (2011) have derived volumetric barrier decomposition algorithms for solving two-stage stochastic semidefinite programs and proved polynomial complexity of certain members of the algorithms. In this paper, we utilize their work to derive volumetric barrier decomposition algorithms for solving two-stage stochastic convex quadratic...

Zhao (2001) introduced a logarithmic barrier interior point method for solving two-stage stochastic linear programs with recourse using Benders’ decomposition. This decomposition algorithm and its analysis have been extended by Alzalg and Ariyawansa (2014) for solving two-stage stochastic programs with recourse over all symmetric cones. In this pap...

Second-order cone programs are a class of convex optimization problems. We refer to them as deterministic second-order cone programs (DSCOPs) since data defining them are deterministic. In DSOCPs we minimize a linear objective function over the intersection of an affine set and a product of second-order (Lorentz) cones. Stochastic programs have bee...

In semidefinite programming (SDP) problems, the variable is a symmetric matrix that is required to be positive semidefinite. Within this chapter, readers delve into the world of SDP, study some combinatorial applications of SDP, and explore the concept of SDP duality, shedding light on the intriguing interplay between primal and dual formulations....

Supported by rigorous math and computer science foundations, Combinatorial and Algorithmic Mathematics: From Foundation to Optimization provides a from-scratch understanding to the field of optimization, discussing 70 algorithms with roughly 220 illustrative examples, 160 nontrivial end-of-chapter exercises with complete solutions to ensure readers...

This chapter provides a comprehensive introduction to fundamental branches of mathematical logic that form the basis for reasoning in mathematics, computer science, and philosophy. More precisely, it serves as a valuable resource for anyone seeking a firm grasp of propositional and predicate logic, from their foundational concepts to practical appl...

This chapter introduces some set-theoretic concepts that serve as the basis for various mathematical structures and algorithms in computer science. More specifically, the chapter covers mathematical induction, sets with their properties, relations, equivalence relations, ordering relations, partition, and functions with their characteristics. The m...

This chapter equips the reader with the essential tools and knowledge needed for counting and effective enumeration. It offers a comprehensive exploration of fundamental concepts in counting. More specifically, we introduce Binomial coefficients and their manipulation and identities. Then we delve into the principles of counting, gaining insight in...

This chapter provides core mathematical concepts that underpin diverse areas, including combinatroics and optimization. It provides essential insights into some foundational analytic and algebraic topics, including sequences, series, matrices, subspaces, bases, polydedra, and convex cones. We also present Farkas' lemma, which is a crucial result in...

This chapter serves as a comprehensive introduction to essential concepts in graph theory, providing a foundation for both theoretical understanding and practical problem-solving in diverse fields. More precisely, it delves into the world of graphs and covers key topics such as graph properties, graph coloring, and directed graphs. Readers will als...

In this chapter, we explore the essential concepts of the analysis of algorithms. By delving into asymptotic notation, readers will gain a deep understanding of how to quantify and assess the running time of algorithms, enabling them to evaluate and compare the efficiency of algorithms, ultimately facilitating the selection of efficient algorithms...

In second-order cone programming (SOCP) problems, we optimize a linear objective function over the intersection of an affine set and the product of second-order cones. Within this chapter, we explore the SOCP problems and their associated interior-point methods, providing a comprehensive understanding of this important class of optimization problem...

Recurrences are equations or inequalities that are used to describe function in terms of their value on smaller inputs. In this chapter, we learn some recurrence-solving techniques, namely guess-and-confirm, recursion-iteration, generating functions, and recursion-tree. This chapter concludes with a set of exercises to encourage readers to apply th...

This chapter describes and analyzes some fundamental algorithms that lie at the core of computer science and numerical analysis. More precisely, it covers some searching and sorting algorithms, namely linear search, binary search, insertion sort, selection sort, and merge sort, which are essential for data manipulation and organization. It also del...

Within this chapter, we introduce fundamental algorithms designed for the purpose of graph exploration and search. Our primary emphasis is placed on two key searching algorithms: The breadth-first search and depth-first search. We also describe the applications of these algorithms in computing spanning trees, computing shortest paths, testing bipar...

In linear programming (LP) problems, we optimize a linear function subject to linear equality and inequality constraints. Within this chapter, we initiate our study of LP, commencing with the graphical method. We delve into the intricacies of LP geometry. Subsequently, we shift our focus to the study of the simplex method, the most prevalent algori...

This is the SOLUTIONS MANUAL for Combinatorial and Algorithmic Mathematics: From Foundation to Optimization 1st edition By Baha Alzalg

In this paper, we present a polynomial-time barrier algorithm for solving multi-stage stochastic convex semidefinite optimization based on the Lagrangian dual method which relaxes the nonanticipativity constraints. We show that the barrier Lagrangian dual functions for our setting form self-concordant families with respect to barrier parameters. We...

This paper explores the nonconvex second-order cone as a nonconvex conic extension of the known convex second-order cone in optimization, as well as a higher-dimensional conic extension of the known causality cone in relativity. The nonconvex second-order cone can be used to reformulate nonconvex quadratic programming and nonconvex quadratically co...

Infinite-dimensional stochastic second-order cone programming involves minimizing linear functions over intersections of affine linear manifolds with infinite-dimensional second-order cones. However, even though there is a legitimate necessity to explore these methods in general spaces, there is an absence of infinite-dimensional counterparts for t...

One of the chief attractions of stochastic mixed-integer second-order cone programming is its diverse applications, especially in engineering (Alzalg and Alioui, IEEE Access, 10:3522-3547, 2022). The linear and nonlinear versions of this class of optimization problems are still unsolved yet. In this paper, we develop a hybrid optimization algorithm...

In this paper, we look into the rotated quadratic cone and analyze its algebraic structure. We construct an algebra associated with this cone and show that this algebra is a Euclidean Jordan algebra (EJA) with a certain inner product. We also demonstrate some spectral and algebraic characteristics of this EJA. The rotated quadratic cone is then pro...

This dissertation explores the nonconvex second-order cone as a nonconvex extension of the known convex second-order cone in mathematics, as well as a higher-dimensional extension of the Einstein-Minkowski causality cone in relativity. The cone can be used to reformulate nonconvex quadratic programming in conic format and can arise in real applicat...

We study the two-stage stochastic infinity norm optimization problem with recourse based on the Jordan algebra. First, we explore and develop the Jordan algebra structure of the infinity norm cone, and utilize it to compute the derivatives of the barrier recourse functions. Then, we prove that the barrier recourse functions and the composite barrie...

In this paper, our central notions of analysis are convexity and betweenness. These concepts are given from the viewpoint of conceptual spaces and are specified in relation to the coordinates of a polar system. We introduce and study a problem that seeks to optimize a polar convex objective subject to polar convex constraints, which we refer to as...

We propose logarithmic-barrier decomposition-based interior-point algorithms for solving two-stage stochastic quadratically constrained convex quadratic programming problems in a Hilbert space. We prove the polynomial complexity of the proposed algorithms, and show that this complexity is independent on the choice of the Hilbert space, and hence it...

All earlier work on optimization problems over the rotated quadratic cones has formulated them as second-order cone programming problems. While doing this can be easier than developing special-purpose algorithms for solving this class of optimization problems, this approach may not be always the cheapest one in terms of computational cost. In this...

In this thesis, we propose and analyze three polynomial-time interior-point algorithms for solving three different classes of stochastic conic programs. In the first part, we develop a volumetric barrier cutting plane algorithm for two-stage stochastic linear semi-infinite programming and present its complexity analysis. The dominant terms in the c...

Second-order cone programming problems are a tractable subclass of convex optimization problems that can be solved using polynomial algorithms. In the last decade, stochastic second-order cone programming problems have been studied, and efficient algorithms for solving them have been developed. The mixed-integer version of these problems is a new c...

Mehrotra and Özevin [SIAM J Optim 19: 1846--1880, 2009] computationally found that a weighted barrier decomposition algorithm for two-stage stochastic conic programs achieves significantly superior performance when compared to standard barrier decomposition algorithms existing in the literature. Inspired by this motivation, the same authors [SIAM J...

We present approximate solutions for the robust semi-infinite multi-objective convex symmetric cone programming problem. By using the robust optimization approach, we establish an approximate optimality theorem and approximate duality theorems for approximate solutions in convex symmetric cone optimization problem involving infinitely many constrai...

We consider a stochastic convex optimization problem over nonsymmetric cones with discrete support. This class of optimization problems has not been studied yet. By using a logarithmically homogeneous self-concordant barrier function, we present a homogeneous predictor-corrector interior-point algorithm for solving stochastic nonsymmetric conic opt...

In this paper, we study a generalized scale-invariant analogue of the well-known Korteweg–de Vries (KdV) equation. This generalized equation can be thought of as a bridge between the KdV equation and the SIdV equation that was discovered recently, and shares the same one-soliton solution as the KdV equation. By employing the auxiliary equation meth...

Second-order cone programming problems are a tractable subclass of convex optimization problems that can be solved using polynomial algorithms. In the last decade, stochastic second-order cone programming problems have been studied, and efficient algorithms for solving them have been developed. The mixed-integer version of these problems is a new c...

We present a logarithmic barrier interior-point method for solving a second-order cone programming problem. Newton’s method is used to compute the descent direction. The main contribution of this paper is that it uniquely uses the so-called majorant functions as an efficient alternative to line search methods to determine the displacement step alon...

In this paper, we study the two-stage stochastic linear semi-infinite programming with recourse to handle uncertainty in data defining (deterministic) linear semi-infinite programming. We develop and analyze volumetric barrier cutting plane interior point methods for solving this class of optimization problems, and present a complexity analysis of...

Several logarithmic-barrier interior-point methods are now available for solving two-stage stochastic optimization problems with recourse in the finite-dimensional setting. However, despite the genuine need for studying such methods in general spaces, there are no infinite-dimensional analogs of these methods. Inspired by this evident gap in the li...

Alzalg (J Optim Theory Appl 163(1):148–164, 2014) derived a homogeneous self-dual algorithm for stochastic second-order cone programs with finite event space. In this paper, we derive an infeasible interior-point algorithm for the same stochastic optimization problem by utilizing the work of Rangarajan (SIAM J Optim 16(4), 1211–1229, 2006) for dete...

In this paper, we develop a primal-dual central trajectory interior-point algorithm for symmetric programming problems and establish its complexity analysis. The main contribution of the paper is that it uniquely equips the central trajectory algorithm with various selections of the displacement step while solving symmetric programming. To show the...

We study and solve the two-stage stochastic extended second-order cone programming problem. We show that the barrier recourse functions and the composite barrier functions for this optimization problem are self-concordant families with respect to barrier parameters. These results are used to develop primal decomposition-based interior-point algorit...

The two-stage stochastic second-order cone program (SSOCP) has been recently introduced in [1] to cover a lot of important applications that cannot be captured by the two-stage stochastic linear program (SLP). Wets [9] described and characterized the equivalent convex program of a two stage SLP. There is no work discussing the equivalent convex pro...

In this paper, we consider the SEIR (Susceptible-Exposed-Infected-Recovered) epidemic model by taking into account both standard and bilinear incidence rates of fractional order. First, the non-negative solution of the SEIR model of fractional order is presented. Then, the multi-step generalized differential transform method (MSGDTM) is employed to...

In elliptic cone optimization problems, we minimize a linear objective function over the intersection of an affine linear manifold with the Cartesian product of the so-called elliptic cones. We present some general classes of optimization problems that can be cast as elliptic cone programmes such as second-order cone programmes and circular cone pr...

Circular programming problems are a new class of convex optimization problems that include second-order cone programming problems as a special case. Alizadeh and Goldfarb [Math. Program. Ser. A 95 (2003) 3--51] introduced primal-dual path-following algorithms for solving second-order cone programming problems. In this paper, we generalize their wor...

Homogeneous self-dual algorithms for stochastic semidefinite programs with finite event space have been proposed by Jin et al. in [17]. Alzalg [8] has adopted their work to derive homogeneous selfdual algorithms for stochastic second-order programs with finite event space. In this paper, we generalize these to derive homogeneous selfdual algorithms...

In this paper, we study and analyze the algebraic structure of the circular cone. We establish a new efficient spectral decomposition, set up the Jordan algebra associated with the circular cone, and prove that this algebra forms a Euclidean Jordan algebra with a certain inner product. We then show that the cone of squares of this Euclidean Jordan...

[This corrects the article DOI: 10.1371/journal.pone.0103617.].

In this work, we consider giving up smoking dynamic on adolescent nicotine dependence. First, we use the Caputo derivative to develop the model in fractional order. Then we apply two different numerical methods to compute accurate approximate solutions of this new model in fractional order and compare their results. In order to do this, we consider...

The optimal power flow problem is concerned with finding a proper operating point for a power network while attempting to minimize a cost function and satisfy network constraints. We analyze the optimal power flow problem subject to contingency constraints and investigate the relationship between the cost of the optimal power flow problem and netwo...

In deterministic mixed integer second-order cone programs (DMISCOPs) we minimizea linear objective function over the intersection of an affine set and a product of second-order (Lorentz) cones, and an additional constraint that requires a subset of the variables attain integers val-ues. We refer to them as deterministic mixed integer second-order c...

Abstract. The optimal power flow problem is concerned with finding a proper operating point for a power network while attempting to minimize some cost function and satisfy several network constraints. In this report we analyze the optimal power flow problem subject to contingency constraints, which demands that there be enough power to meet demands...

For an integer k>1, a cycle-complete graph Smarandache-Ramsey number r s k (C m ,K n ) is the smallest integer N such that every graph G of order N contains k cycles, C m , on m vertices or the complement of G contains k complete graph, K n , on n vertices. If k=1, then the Smarandache-Ramsey number r s k (C m ,K n ) is nothing but the classical Ra...

The cycle-complete graph Ramsey number r(C m, K n) is the smallest integer N such that every graph G of order N contains a cycle C m on m vertices or has independent number α(G) ≥ n. It has been conjectured by Erdos, Faudree, Rousseau and Schelp that r(C m, K n) = (m -1)(n -1) + 1 for all m ≥ n ≥ 3 (except r(C 3, K 3) = 6). In this paper, we show t...

The cycle-complete graph Ramsey number r(C m, K n) is the smallest integer N such that every graph G of order N contains a cycle C m on m vertices or has independent number α(G) ≥ n. It has been conjectured by Erdös, Faudree, Rousseau and Schelp that r(C m, K n) = (m - 1)(n - 1) + 1 for all m ≥ n ≥ 3 (except r(C 3, K 3) = 6). In this paper we will...