# Optimization Over Integers.pdfl

The purpose of this book is to provide a unified, insightful, and modern treatment of the theory of integer optimization with an eye towards the future. We have selected those topics that we feel have influenced the current state of the art and most importantly we feel will affect the future of the field. We depart from earlier treatments of integer optimization by placing significant emphasis on strong formulations, duality, algebra and most importantly geometry.

## Optimization Over Integers.pdfl

We consider robust counterparts of integer programs and combinatorial optimization problems (summarized as integer problems in the following), i.e., seek solutions that stay feasible if at most Γ-many parameters change within a given range. While there is an elaborate machinery for continuous robust optimization problems, results on robust integer problems are still rare and hardly general.

We show several optimization and approximation results for the robust (with respect to cost, or few constraints) counterpart of an integer problem under the condition that one can optimize or approximate the original integer problem with respect to a piecewise linear objective (respectively piecewise linear constraints).

We demonstrate the applicability of our approach on two classes of integer programs, namely, totally unimodular integer programs and integer programs with two variables per inequality. Further, for combinatorial optimization problems our method yields polynomial time approximations and pseudopolynomial, exact algorithms for Robust Unbounded Knapsack Problems.

Integer programming algorithms minimize or maximize a function subject to equality, inequality, and integer constraints. Integer constraints restrict some or all of the variables in the optimization problem to take on only integer values. This enables accurate modeling of problems involving discrete quantities (such as shares of a stock or cells in a battery) or yes-or-no decisions. When there are integer constraints on only some of the variables, the problem is called a mixed-integer program (MIP). Example integer programming problems include portfolio optimization in finance, optimal dispatch of generating units (unit commitment) in energy production, design optimization in engineering, and scheduling and routing in transportation and supply chain applications.

See also:Optimization Toolbox, Global Optimization Toolbox, linear programming, quadratic programming, nonlinear programming, genetic algorithm, investment management, energy trading, prescriptive analytics, surrogate optimization, power system simulation and optimization

Mixed-Integer Linear Optimization has been an important topic in optimization theory and applications since the 1960s. As a mathematical subject, it is a rich combination of aspects of geometry, algebra, number theory, and combinatorics. The interplay between the mathematics, modeling, and algorithmics makes it a deep and fascinating subject of applied mathematics, which has had an enormous impact on real-world applications. But many physical systems have nonlinear aspects and further discrete design aspects. So we are naturally led to the paradigm of Mixed-Integer Non-Linear Optimization. But the mathematics and effective algorithmics of this subject are far more daunting than the linear case, and so there is a focus on broad sub-classes where results from the linear world can be lifted up. Furthermore, effective modeling techniques are much more subtle and are intertwined with state-of-the-art algorithmics and software which are rapidly evolving.

A famous and classical example of a problem in discrete optimization is the traveling salesperson problem: For given cities and distances of traveling from one city to another, we seek to find the shortest route that visits each city once and returns to the starting city. Discrete optimization problems naturally arise in many kinds of applications including bioinformatics, telecommunications network design, airline scheduling, circuit design, and efficient resource allocation. The field also connects to a variety of areas in mathematics, computer science, and data analytics including approximation algorithms, convex and tropical geometry, number theory, real algebraic geometry, parameterized complexity theory, quantum computing, machine learning, and mathematical logic.

This Boot-camp will be the opening event of the semester and it aims to attract young researchers to this topic.The four mini courses, presented by four speakers known for high-quality exposition, will cover various subjects such as new advances in approximation algorithms, mixed integer non-linear programming, algebraic techniques in optimization and applications to social sciences. The event provides a taste of the many methods and hot topics to be discussed during the semester. The event will also include a poster session to allow graduate students to present their work and other community building activities.

Combinatorial optimization is an active research field in mathematics, with an immense range of applications. This workshop will bring together researchers and leading experts interested in the mathematical foundations of combinatorial optimization algorithms to discuss new tools and methods, in particular regarding the use ofalgebraic, analytical, and geometric techniques. Special emphasis will be given on polyhedral methods, since they are at the core of several groundbreaking combinatorial optimization results developed in recent years.

The aim of this workshop is to discuss many exciting recent developments on the computational side of discrete optimization. The workshop has three main themes. The first theme is that of commercial and academic/open-source solvers that have allowed the solution of very large-scale problems, and of recent developments in exact solvers that have allowed for proofs of results in logic, knot theory, and combinatorics. The second theme is the interaction between optimization and machine learning: these two areas complement each other in several ways. The third theme is quantum computing and unconventional computing architectures: quantum computing has been used to tackle combinatorial optimization problems, and quantum algorithms exist for other related optimization problems such as linear and semidefinite relaxations.

Quantum annealing, as the core algorithm of a D-Wave quantum computer, has the potential to approach or even achieve the global optima in an exponential solution space, corresponding to the quantum evolution towards the ground state of the Hamiltonian problem24. The quantum processing units (QPUs), which are the core components for performing quantum annealing, are designed to solve quadratic unconstrained binary optimization (QUBO) problems25,26, where each qubit represents a variable, and the couplers between qubits represent the costs associated with qubit pairs.

Quantum annealing uses the quantum effects generated by quantum fluctuations to realize the global optimal solution of the objective function. The integer factorization problem can be transformed into a combination optimization problem that can be handled by the quantum annealing algorithm, and the minimum energy value can be output through the quantum annealing algorithm. At this time, the minimum value is the successful solution of integer factorization. To clarify the integer factorization method via quantum annealing, we introduce a multiplication table to illustrate the feasibility of mapping the integer factorization problem to Ising model (a model can be processed by a D-Wave quantum computer). We illustrate the factorization of the integer multiplication table by factoring \(N=p\times q\), where \(p\) and \(q\) are prime numbers. Table 1 shows the factorization of \(143=11\times 13\). In Table 1, \(p_i\) and \(q_i\) represent the bits of the multipliers, and \(z_ij\) is the carried bits from \(i\)th bit to the \(j\)th bit. All the variables \(p_i\), \(q_i\), and \(z_ij\) in the equations are binary.

As mentioned above, we mainly focus on the optimization of the model parameters. Jiang et al.30 a way to reduce the 3-local term to a 2-local term, which increased the local field coefficient and coupler strength parameters, especially for large integers. In the integer factorization problem based on quantum annealing, the reduction of the model parameters is beneficial to reducing the hardware requirements and the precision of quantum annealing. To reduce the 3-local term to a 2-local term in the integer factorization process, inspired by ref. 35, we optimize Eq. (20) of ref. 32 and form a new dimension reduction method from the 3-local term to 2-local term, as shown in Eq. (26)

The negative term \(-x_1x_2x_3=-\mathop\min \limits_x_4(x_4x_3+2(x_1x_2-2x_1x_4-2x_2x_4+3x_4))\) is the same as ref. 30. We mainly prove our optimization of the positive term, that is, why the positive term \(x_1x_2x_3=\mathop\min \limits_x_4(x_4x_3+x_1x_2-x_1x_4\) \(-x_2x_4+x_4)\) holds.

Tables 4 and 5 show that the optimization model can further reduce the weight of the qubits and the range of the coupler strength involved in the problem model, which can advance the large-scale integers in the D-Wave machine.

An integer programming problem is a mathematical optimization or feasibility program in which some or all of the variables are restricted to be integers. In many settings the term refers to integer linear programming (ILP), in which the objective function and the constraints (other than the integer constraints) are linear.

The feasible integer points are shown in red, and the red dashed lines indicate their convex hull, which is the smallest convex polyhedron that contains all of these points. The blue lines together with the coordinate axes define the polyhedron of the LP relaxation, which is given by the inequalities without the integrality constraint. The goal of the optimization is to move the black dashed line as far upward while still touching the polyhedron. The optimal solutions of the integer problem are the points ( 1 , 2 ) \displaystyle (1,2) and ( 2 , 2 ) \displaystyle (2,2) which both have an objective value of 2. The unique optimum of the relaxation is ( 1.8 , 2.8 ) \displaystyle (1.8,2.8) with objective value of 2.8. If the solution of the relaxation is rounded to the nearest integers, it is not feasible for the ILP.