A primal–dual interior point algorithm with an exact and differentiable merit function for nonlinear programming

Validation of measurement data using an interior point SQP

In this chapter, an implementation of an interior point filter line-search algorithm for large-scale nonlinear programming proposed by Wachter and Biegler (2005a, b) is presented. As we know, to allow convergence from poor starting points and to enforce progress to the solution, interior point methods both in trust region and in line-search frameworks with exact penalty merit function have been developed. For example, KNITRO uses the l 1 exact penalty function (Byrd, Hribar, & Nocedal, 1999; Byrd, Gilbert, & Nocedal, 2000). On the other hand, Fletcher and Leyffer (2002) proposed filter methods as an alternative to merit functions, as a tool for global convergence guarantee in algorithms for nonlinear optimization. The idea of filter is that trial points are accepted if they improve the objective function value or improve the constraint violation instead of a combination of these two measures defined by the merit function. Even if the filter methods include different heuristics, they have been adapted to barrier methods in a number of ways. For example, Ulbrich et al. (2004) considered a trust-region filter method and accept the trial step on the basis of the norm of the optimality conditions. Benson et al. (2002a) proposed several heuristics using the idea of filter methods, for which improved efficiency is reported compared to their previous merit function approach. The global convergence of an interior point algorithm with a filter line search was analyzed by Wachter and Biegler (2001). Here, the assumptions made for the analysis of the global convergence are less restrictive than those made for line-search interior point methods for nonlinear programming developed, for example, by El-Bakry et al. (1996), Yamashita (1998), or Tits et al. (2003).

Based on a new parametric kernel function, this paper presents a primal-dual large-update interior-point algorithm (IPM) for semi-definite optimization (SDO) problems. The new parametric function is neither self-regular function nor the usual logarithmic barrier function. It is strongly convex and possesses some novel analytic properties. We analyse this new parametric kernel function and show that the proposed algorithm has favorable complexity bound in terms of the analytic properties of the kernel function. Moreover, the complexity bound for our large-update IPM is shown to be O(\sqrt{n}(\log n)^2 \log\frac{n}{\epsilon}). Some numerical results are reported to illustrate the feasibility of the proposed algorithm.

We present a primal-dual interior-point algorithm for solving optimization problems with nonlinear inequality constraints. The algorithm has some of the theoretical properties of trust region methods, but works entirely by line search. Global convergence properties are derived without assuming regularity conditions. The penalty parameter $\rho$ in the merit function is updated adaptively and plays two roles in the algorithm. First, it guarantees that the search directions are descent directions of the updated merit function. Second, it helps to determine a suitable search direction in a decomposed SQP step. It is shown that if $\rho$ is bounded for each barrier parameter $\mu$, then every limit point of the sequence generated by the algorithm is a Karush--Kuhn--Tucker point, whereas if $\rho$ is unbounded for some $\mu$, then the sequence has a limit point which is either a Fritz--John point or a stationary point of a function measuring the violation of the constraints. Numerical results confirm that the algorithm produces the correct results for some hard problems, including the example provided by Wächter and Biegler, for which many of the existing line search--based interior-point methods have failed to find the right answers.

The traditional load-shedding models usually use global optimization to get the load-shedding region, which will cause multiple variables, huge computing scale and other problems. This makes it hard to meet the requirements of timeliness in on-line power system operation risk assessment. In order to solve the problems of the present load-shedding models, a load-shedding model based on sensitivity analysis is proposed in this manuscript. By calculating the sensitivity of each branch on each bus, the collection of buses which have remarkable influence on reducing the power flow on over-load branches is obtained. In this way, global optimization is turned to local optimization, which can narrow the solution range. By comprehensively considering the importance of load bus and adjacency principle regarding the electrical coupling relationship, a load-shedding model is established to get the minimum value of the load reduction from different kinds of load buses, which is solved by the primal dual interior point algorithm. In the end, different cases on the IEEE 24-bus, IEEE 300-bus and other multi-node systems are simulated. The correctness and effectiveness of the proposed load-shedding model are demonstrated by the simulation results.

A new nonmonotone line-search procedure is presented for the generally constrained case of nonlinear programming problems. The new algorithm is based on the use of standard penalty methods for the definition of merit functions used during line search to find the next iterate in algorithms generating a search direction iteratively. The key concept is the discretization of the penalty parameter used over a finite range of orders of magnitude and the provision of a memory list for each such order, as in standard nonmonotone line-search procedures used for unconstrained optimization. Nonmonotonicity helps in escaping from local minima, while the discretized penalty parameters overcome the difficulties in choosing a penalty parameter that varies, but having the same definition as the problem while not underpenalizing the constraints to arrive at the desired KKT point. An implementation within a customized logarithmic barrier algorithm for bounds' handling is presented with capabilities for very large scale applications; the algorithm uses exact first and second derivative information, derived symbolically, and the search direction is generated by solution of the Lagrange−Newton equations. The case studies presented demonstrate the capabilities of the new line-search procedure, and comparisons with other methods are discussed. It is noted that we found a significantly better solution in case study 5. The new nonmonotone line-search procedure is, at present, a heuristic and from the computational point of view: future work will focus on the investigation of both the theoretical properties of the method and new implementation aspects.

Kernel functions play an important role in defining new search directions for primal-dual interior-point algorithm. In this paper, a new kernel function which its barrier term is integral type is proposed. We study the properties of the new kernel function, and give a primal-dual interior-point algorithm for solving linear optimization based on the new kernel function. Polynomial complexity of algorithm is analyzed. The iteration bounds both for large-update and for small-update methods are obtained, respectively. The iteration bound for small-update method is the best known complexity bound.

This addition to the ISOR series introduces complementarity models in a straightforward and approachable manner and uses them to carry out an in-depth analysis of energy markets, including formulation issues and solution techniques. In a nutshell, complementarity models generalize: a. optimization problems via their Karush-Kuhn-Tucker conditions b. on-cooperative games in which each player may be solving a separate but related optimization problem with potentially overall system constraints (e.g., market-clearing conditions) c. conomic and engineering problems that arent specifically derived from optimization problems (e.g., spatial price equilibria) d. roblems in which both primal and dual variables (prices) appear in the original formulation (e.g., The National Energy Modeling System (NEMS) or its precursor, PIES). As such, complementarity models are a very general and flexible modeling format. A natural question is why concentrate on energy markets for this complementarity approach? s it turns out, energy or other markets that have game theoretic aspects are best modeled by complementarity problems. The reason is that the traditional perfect competition approach no longer applies due to deregulation and restructuring of these markets and thus the corresponding optimization problems may no longer hold. Also, in some instances it is important in the original model formulation to involve both primal variables (e.g., production) as well as dual variables (e.g., market prices) for public and private sector energy planning. Traditional optimization problems can not directly handle this mixing of primal and dual variables but complementarity models can and this makes them all that more effective for decision-makers.

Kernel functions serve the central goal of creating new search directions for the primal-dual interiorpoint algorithm to solve linear optimization problems. A significantly improved primal-dual interior-point algorithm for linear optimization is presented based on a novel kernel function. We show a primal-dual interior-point technique for linear optimization based on a class of kernel functions that are eligible. This research presents a new efficient kernel function-based primal-dual IPM algorithm for semidefinite programming problems based on the Nesterov-Todd (NT) direction. With a new and simple technique, we propose a new kernel function to obtain an optimal solution of the perturbed problem (SDP)µ. We obtain the best-known complexity results,for smalland large-update namely O(pp+1/2p √nlog tr(X0S0)/ε) and O((pn)p+1/2p log trX0S0/ε) large update To prove the effectiveness of our proposed kernel function, we compare our numerical results with some alternatives presented by Touil et al. (2017).

We present a unified analysis for a class of long-step primal-dual path-following algorithms for semidefinite programming whose search directions are obtained through linearization of the symmetrized equation of the central path Hp(XS) -- [PXSP -~ + (PXSP 1)TI/2 = #I, introduced by Zhang. At an iterate (X, S), we choose a scaling matrix P from the class of nonsingular matrices P such that PXSP -~ is symmetric. This class of matrices includes the three well-known choices, namely: P = S ~/2 and P - X -~/2 proposed by Monteiro, and the matrix P corresponding to the Nesterov Todd direction. We show that within the class of algorithms studied in this paper, the one based on the Nesterov-Todd direction has the lowest possible iteration-complexity bound that can provably be derived from our analysis. More specifically, its iteration-complexity bound is of the same order as that of the corresponding long-step primal-dual path-following algorithm for linear programming introduced by Kojima, Mizuno and Yoshise. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.

In this paper we propose a new class of primal-dual path-following interior point algorithms for solving monotone linear complementarity problems. At each iteration, the method would select a target on the central path with a large update from the current iterate, and then the Newton method is used to get the search directions, followed by adaptively choosing the step sizes, which are, e.g., the largest possible steps before leaving a neighborhood that is as wide as the given ${\cal N}^-_{\infty}$ neighborhood. The only deviation from the classical approach is that we treat the classical Newton direction as the sum of two other directions, corresponding to, respectively, the negative part and the positive part of the right-hand side. We show that if these two directions are equipped with different and appropriate step sizes, then the method enjoys the low iteration bound of $O(\sqrt{n}\log L)$, where n is the dimension of the problem and $L=\frac{(x^0)^Ts^0}{\ep}$ with $\ep$ the required precision and $(x^0,s^0)$ the initial interior solution. For a predictor-corrector variant of the method, we further prove that, besides the predictor steps, each corrector step also reduces the duality gap by a rate of $1-1/O(\sqrt{n})$. Additionally, if the problem has a strict complementary solution, then the predictor steps converge Q-quadratically.

In this paper a primal-dual path-following interior-point algorithm for the monotone semidefinite linear complementarity problem is presented. The algorithm is based on Nesterov-Todd search directions and on a suitable proximity for tracing approximately the central-path. We provide an unified analysis for both long and small-update primal-dual algorithms. Finally, the iteration bounds for these algorithms are obtained.

This paper presents a new primal-dual interior-point algorithm with reduced potential function for a class of convex programming, based on the ideas of that method for solving linear programming. The new algorithm chooses the classical Newton direction as iteration direction and its iteration stepsize is determined by potential function. As the search directions Δxand Δsaren't orthogonal any more, the complexity analysis of this method is different from that of linear programming, correspondingly. Under a scaled Lipschitz condition, the algorithm is proved to possess O(nL) iteration-complexity bounds

: A globally convergent primal-dual predictor corrector interior point algorithm for linear programming has been developed.

We present primal–dual interior-point algorithms with polynomial iteration bounds to find approximate solutions of semidefinite programming problems. Our algorithms achieve the current best iteration bounds and, in every iteration of our algorithms, primal and dual objective values are strictly improved.