A Method for Linearly Constrained Minimization Problems

This paper presents a method, called the convex simplex method, for minimizing a convex objective function subject to linear inequality constraints. The method is a true generalization of Dantzig's linear simplex method both in spirit and in the fact that the same tableau and variable selection techniques are used. With a linear objective function the convex simplex method reduces to the linear simplex method. Moreover, the convex simplex method actually behaves like the linear simplex method whenever it encounters a linear portion of a convex objective function. Many of the sophisticated techniques designed to enhance the efficiency of the linear simplex method are applicable to the convex simplex method. In particular, as an example, a network transportation problem with a convex objective function is solved by using the standard transportation tableau and by only slightly modifying the usual procedure for a linear objective function.

We solve multiperiod nonlinear production smoothing problems by combining the convex simplex method due to Zangwill and the candidate constraint method of Sethi and Thompson. We show that, for the solved problems of size up to 120 constraints and 84 variables, computational savings of 50 percent to 60 percent over the convex simplex method can be realized by combining it with the candidate constraint method. The saving is mainly due to the reduction in the effective size of the problem and in the number of iterations performed. The method is also shown to be applicable to a wide range of problems besides production smoothing.

The problem of obtaining an optimal spline with free knots is tantamount to minimizing derivatives of a nonlinear differentiable function over a Banach space on a compact set. While the problem of data interpolation by quadratic splines has been accomplished, interpolation by splines of higher orders is far more challenging. In this paper, to overcome difficulties associated with the complexity of the interpolation problem, the interval over which data points are defined is discretized and continuous derivatives are replaced by their discrete counterparts. The l∞-norm used for maximum rth order curvature (a derivative of order r) is then linearized, and the problem to obtain a near-optimal spline becomes a linear programming (LP) problem, which is solved in polynomial time by using LP methods, e.g., by using the Simplex method implemented in modern software such as CPLEX. It is shown that, as the mesh of the discretization approaches zero, a resulting near-optimal spline approaches an optimal spline. Splines with the desired accuracy can be obtained by choosing an appropriately fine mesh of the discretization. By using cubic splines as an example, numerical results demonstrate that the linear programming (LP) formulation, resulting from the discretization of the interpolation problem, can be solved by linear solvers with high computational efficiency and the resulting spline provides a good approximation to the sought-for optimal spline.

For many of us, modern-day linear programming (LP) started with the work of George Dantzig in 1947. However, it must be said that many other scientists have also made seminal contributions to the subject, and some would argue that the origins of LP predate Dantzig’s contribution. It is matter open to debate [36]. However, what is not open to debate is Dantzig’s key contribution to LP computation. In contrast to the economists of his time, Dantzig viewed LP not just as a qualitative tool in the analysis of economic phenomena, but as a method that could be used to compute actual answers to specific real-world problems. Consistent with that view, he proposed an algorithm for solving LPs, the simplex algorithm [12]. To this day the simplex algorithm remains a primary computational tool in linear and mixed-integer programming (MIP). In [11] it is reported that the first application of Dantzig’s simplex algorithm to the solution of a non-trivial LP was Laderman’s solution of a 21 constraint, 77 variable instance of the classical Stigler Diet Problem [41]. It is reported that the total computation time was 120 man-days! The first computer implementation of an at-least modestly general version of the simplex algorithm is reported to have been on the SEAC computer at the then National Bureau of Standards [25]. (There were apparently some slightly earlier implementations for dealing with models that were “triangular”, that is, where all the linear systems could be solved by simple addition and subtraction.) Orchard-Hays [35] reports that several small instances having as many as 10 constraints and 20 variables were solved with this implementation. The first systematic development of computer codes for the simplex algorithm began very shortly thereafter at the RAND Corporation in Santa Monica, California. Dantzig’s initial LP work occurred at the Air Force following

In this paper the problem of finding the error of an approximate solution for ill-posed problems on some compact sets is discussed. Sets of bounded monotonic or convex functions and sets of functions with a given Lipschitz constant are considered. These functions are given on a segment. Two approaches are used to construct sets of approximate solutions. In the first approach we assume that a function for the right-hand side of an operator equation is given on the whole segment. In the second approach we use only grid values of this function. Applying these approaches, we obtain quadratic or linear programming problems and find functions bounding above and below these approximate solutions. In the paper we use two algorithms to find errors that determine the sets of approximate solutions for the cases when operators of the problems are positive or integral. The inverse problem for the heat conduction equation is considered as a model example.

Secure multiparty computation is a basic concept of growing interest in modern cryptography. It allows a set of mutually distrusting parties to perform a computation on their private information in such a way that as little as possible is revealed about each private input. The early results of multiparty computation have only theoretical signi??cance since they are not able to solve computationally complex functions in a reasonable amount of time. Nowadays, e??ciency of secure multiparty computation is an important topic of cryptographic research. As a case study we apply multiparty computation to solve the problem of secure linear programming. The results enable, for example in the context of the EU-FP7 project SecureSCM, collaborative supply chain management. Collaborative supply chain management is about the optimization of the supply and demand con??guration of a supply chain. In order to optimize the total bene??t of the entire chain, parties should collaborate by pooling their sensitive data. With the focus on e??ciency we design protocols that securely solve any linear program using the simplex algorithm. The simplex algorithm is well studied and there are many variants of the simplex algorithm providing a simple and e??cient solution to solving linear programs in practice. However, the cryptographic layer on top of any variant of the simplex algorithm imposes restrictions and new complexity measures. For example, hiding the number of iterations of the simplex algorithm has the consequence that the secure implementations have a worst case number of iterations. Then, since the simplex algorithm has exponentially many iterations in the worst case, the secure implementations have exponentially many iterations in all cases. To give a basis for understanding the restrictions, we review the basic theory behind the simplex algorithm and we provide a set of cryptographic building blocks used to implement secure protocols evaluating basic variants of the simplex algorithm. We show how to balance between privacy and e??ciency; some protocols reveal data about the internal state of the simplex algorithm, such as the number of iterations, in order to improve the expected running times. For the sake of simplicity and e??ciency, the protocols are based on Shamir's secret sharing scheme. We combine and use the results from the literature on secure random number generation, secure circuit evaluation, secure comparison, and secret indexing to construct e??cient building blocks for secure simplex. The solutions for secure linear programming in this thesis can be split into two categories. On the one hand, some protocols evaluate the classical variants of the simplex algorithm in which numbers are truncated, while the other protocols evaluate the variants of the simplex algorithms in which truncation is avoided. On the other hand, the protocols can be separated by the size of the tableaus. Theoretically there is no clear winner that has both the best security properties and the best performance.

Summary form only given, as follows. L/sub 2/-minimization problems are commonly solved using one of the following methods: (i) variants of the simplex method, used to solve the L/sub 1/-minimization problem formulated as a linear programming (LP) problem, and (ii) the iteratively reweighted least-squares (IRLS) method, a method favored in some signal processing applications. Interior-point methods (primal affine and Karmarkar's dual affine methods) are considerably faster than the simplex method for solving large LP problems. The principles of affine algorithms and their implementation strongly resemble the IRLS method. However, an efficient implementation is essential to obtain good performances from the interior-point methods. The implementation details for dense and sparse L/sub 1/-minimization problems with and without linear inequality constraints are discussed. A number of examples are worked out, and comparisons are made with existing algorithms wherever possible.

We consider the following type of minimization problem. Given a real valued function f on an open nonempty subset U of a real Banach space E we are looking for a minimum of f on the subset of U which is determined by the constraint condition \(g(x)=y\) where \(g: U \longrightarrow F\) is a given function on U with values in some Banach space F and \(y\in F\) is some selected point, i.e., we are looking for a minimum of f on the level set \([g=y]=\left\{x \in U; g(x)=y\right\}\) of g. If g is of class \(\mathcal{C}^1\) the level surfaces of g have a proper tangent space at regular points x 0 (i.e., \(g^{\prime}(x_0)\) is surjective) at which the nullspace of \(g^{\prime}(x_0)\) has a topological complement. This can be used to introduce local coordinates on the level surface \([g=y]\) in the neighborhood of such a regular point and the constrained minimization problem becomes a minimization problem on an open subset (in terms of the local coordinates) and the characterization of extremal points as derived earlier applies. The conclusion is that a Lagrange multiplier exists, i.e., a continuous linear mapping \(\ell: F \longrightarrow \mathbb{R}\) such that \(f^{\prime}(x_0)= \ell \circ g^{\prime}(x_0)\). The geometrical meaning of this condition is first explained heuristically in a two dimensional setting. In practical cases of interest the condition of the existence of a topological compliment is always met. The well known finite dimensional case is easily obtained. We conclude by some comments on the constrained minimization problem of Dido.

The stochastic variational inequality (VI) has been used widely in engineering and economics as an effective mathematical model for a number of equilibrium problems involving uncertain data. This paper presents a new expected residual minimization (ERM) formulation for a class of stochastic VI. The objective of the ERM-formulation is Lipschitz continuous and semismooth which helps us guarantee the existence of a solution and convergence of approximation methods. We propose a globally convergent (a.s.) smoothing sample average approximation (SSAA) method to minimize the residual function; this minimization problem is convex for the linear stochastic VI if the expected matrix is positive semidefinite. We show that the ERM problem and its SSAA problems have minimizers in a compact set and any cluster point of minimizers and stationary points of the SSAA problems is a minimizer and a stationary point of the ERM problem (a.s.). Our examples come from applications involving traffic flow problems. We show that the conditions we impose are satisfied and that the solutions, efficiently generated by the SSAA procedure, have desirable properties.

We describe an approach for finding upper bounds on an ODE dynamical system’s maximal Lyapunov exponent (LE) among all trajectories in a specified set. A minimisation problem is formulated whose infimum is equal to the maximal LE, provided that trajectories of interest remain in a compact set. The minimisation is over auxiliary functions that are defined on the state space and subject to a pointwise inequality. In the polynomial case—i.e. when the ODE’s right-hand side is polynomial, the set of interest can be specified by polynomial inequalities or equalities, and auxiliary functions are sought among polynomials—the minimisation can be relaxed into a computationally tractable polynomial optimisation problem subject to sum-of-squares constraints. Enlarging the spaces of polynomials over which auxiliary functions are sought yields optimisation problems of increasing computational cost whose infima converge from above to the maximal LE, at least when the set of interest is compact. For illustration, we carry out such polynomial optimisation computations for two chaotic examples: the Lorenz system and the Hénon–Heiles system. The computed upper bounds converge as polynomial degrees are raised, and in each example we obtain a bound that is sharp to at least five digits. This sharpness is confirmed by finding trajectories whose leading Lyapunov exponents approximately equal the upper bounds.

How is it possible to describe any analytic or harmonic function on a compact set in terms of much simpler “construction blocks” like polynomials? Or, how is it possible to know the values of a function inside a compact domain, by knowing only its values on the boundary? Well, these simplifications are possible because the “bricks” are actually organized in complicated and versatile structures. For example the B ∗ algebras. And in addition, the compact domains are certainly among the simples ones, being always reducible to finite reunions. Actually, a complicated structure like a B ∗ algebra, defined by 24 axioms (out of which 13 axioms on commutative algebras, five axioms on norm, one for completeness, and five more specific axioms) can be realized by continuous functions defined on a compact set. It is not the only example. The space l 1 of complex sequences with norm given by the sum of the modules of the terms is isomorphic with the algebra of functions whose Fourier series is absolutely convergent. Also, a compact Hausdorff space, with topology induced by distance, is homeomorphic with a compact subset of [0, 1]N. Any two separable Hilbert spaces are isomorphic, and so on. These similarities bring a unifying point of view: objects of apparently distinct nature, like Weierstrass–Stone theorem on function approximation, Wiener theorem on absolutely convergent Fourier series, spectral expansion of self-adjoint operators, the theorems of Tikhonov, Stone–Čech, or the fixed-point theorem of Brouwer, the Cauchy formula on complex functions, the Green representation theorem, the Poincaré Lemma, etc. actually provide the same fundamental truth: simplification by approximation is possible on compacts.KeywordsHarmonic FunctionCotangent BundleRepresentation FormulaExterior DerivativeCompact Hausdorff SpaceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

By a constructive method, it is proved that a linear sum of rather a small number of differentiable basis functions can well trace any surface defined by a polynomial in several variables on any compact sets. The linear sum is obtained in an explicit form. When the basis function is sufficiently many times differentiable, the tracing is so well done that the surfaces defined by derivatives of the polynomial can be simultaneously traced. This capability of a smooth basis function is partly based on the fact that it has various features of curvature. The radial basis function in two variables, for example, can trace any kinds of quadratic surfaces. This is different from the case of approximation by sigmoid functions. Since the proofs in this paper are concrete, they can be used as algorithms in applications.

This paper establishes a framework for solving some optimization problems with linear constraints using simplex-type methods. The problems include those found in linear programs, linear fractional programs, and generalized linear fractional programs. In this study, these problems refer to a standard form of minimizing a single parameter subject to parameterized linear equations. Based on the analysis of parameterized basis-based solutions, a unified simplex-type approach is proposed. The adaptability of the parameterized model and that of the solution procedure are discussed. In particular, the proposed algorithm can prevent cycling when compared with the conventional simplex method used for solving linear programs.

Abstract: Linear Programming (LPP) is a mathematical method used to optimize resource allocation systems with linear relationships, it aims to maximize or minimize a linear objective function while adhering to linear constraints. This paper showcase how to optimize linear programming problems through the simplex and graphical methods. The excel environment facilitates efficient modeling and solution generation. The simplex method is executed step – by – step, showcasing its versatility, while the graphical method provides visual insights. The solver add in automates the optimization process, adjusting decision variables within constraints. Comparative analysis reveals insights into the strengths of both the methods used in the process of the study. This research contributes to practical applications of LP, emphasizing the significance of excel and solver for real – world problem solving

An algorithm for finding the shortest element of a polyhedral set with application to lagrangian duality