Mathematical optimization

Mathematical optimization or mathematical programming is the selection of a best element from some set of available alternatives. Optimization problems of sorts arise in all quantitative disciplines from computer science and engineering to operations research and economics, the development of solution methods has been of interest in mathematics for centuries. In the simplest case, an optimization problem consists of maximizing or minimizing a real function by systematically choosing input values from within an allowed set and computing the value of the function; the generalization of optimization theory and techniques to other formulations constitutes a large area of applied mathematics. More optimization includes finding "best available" values of some objective function given a defined domain, including a variety of different types of objective functions and different types of domains. An optimization problem can be represented in the following way: Given: a function f: A → ℝ from some set A to the real numbers Sought: an element x0 ∈ A such that f ≤ f for all x ∈ A or such that f ≥ f for all x ∈ A.

Such a formulation is called a mathematical programming problem. Many real-world and theoretical problems may be modeled in this general framework. Since the following is valid f ≥ f ⇔ f ~ ≤ f ~ with f ~:= − f, f ~: A → R it is more convenient to solve minimization problems. However, the opposite perspective would be valid, too. Problems formulated using this technique in the fields of physics may refer to the technique as energy minimization, speaking of the value of the function f as representing the energy of the system being modeled. In machine learning, it is always necessary to continuously evaluate the quality of a data model by using a cost function where a minimum implies a set of optimal parameters with an optimal error. A is some subset of the Euclidean space ℝn specified by a set of constraints, equalities or inequalities that the members of A have to satisfy; the domain A of f is called the search space or the choice set, while the elements of A are called candidate solutions or feasible solutions.

The function f is called, variously, an objective function, a loss function or cost function, a utility function or fitness function, or, in certain fields, an energy function or energy functional. A feasible solution that minimizes the objective function is called an optimal solution. In mathematics, conventional optimization problems are stated in terms of minimization. A local minimum x* is defined as an element for which there exists some δ > 0 such that ∀ x ∈ A where ‖ x − x ∗ ‖ ≤ δ, the expression f ≤ f holds. Local maxima are defined similarly. While a local minimum is at least as good as any nearby elements, a global minimum is at least as good as every feasible element. Unless the objective function is convex in a minimization problem, there may be several local minima. In a convex problem, if there is a local minimum, interior, it is the global minimum, but a nonconvex problem may have more than one local minimum not all of which need be global minima. A large number of algorithms proposed for solving the nonconvex problems – including the majority of commercially available solvers – are not capable of making a distinction between locally optimal solutions and globally optimal solutions, will treat the former as actual solutions to the original problem.

Global optimization is the branch of applied mathematics and numerical analysis, concerned with the development of deterministic algorithms that are capable of guaranteeing convergence in finite time to the actual optimal solution of a nonconvex problem. Optimization problems are expressed with special notation. Here are some examples: Consider the following notation: min x ∈ R This denotes the minimum value of the objective function x2 + 1, when choosing x from the set of real numbers ℝ; the minimum value in this case is 1, occurring at x = 0. The notation max x ∈ R 2

Bruce Peninsula National Park

Bruce Peninsula National Park is a national park on the Bruce Peninsula in Ontario, Canada. Located on a part of the Niagara Escarpment, the park comprises 156 square kilometres and is one of the largest protected areas in southern Ontario, forming the core of UNESCO's Niagara Escarpment World Biosphere Reserve; the park offers opportunities for many outdoor activities, including hiking and bird watching. The park has trails ranging in difficulty from easy to expert, connects to the Bruce Trail; the park offers visitors vistas to view either the sunrise or sunset, the rocks of the Niagara Escarpment, the wildlife, which includes black bear, many species of birds, wild orchids, massasauga rattlesnake, much more. The park was the subject of a short film in 2011's National Parks Project, directed by Daniel Cockburn and scored by John K. Samson, Christine Fellows and Sandro Perri; the Niagara Escarpment runs from near Rochester, New York, to Tobermory on to Manitoulin, St. Joseph Island and other islands located in northern Lake Huron where it turns westwards into the Upper Peninsula of northern Michigan, south of Sault Ste.

Marie. The escarpment extends southwards into Wisconsin following the Door Peninsula and more inland from the western coast of Lake Michigan and Milwaukee ending northwest of Chicago near the Wisconsin-Illinois border, it forms the backbone of the Bruce Peninsula and shapes the northern boundary of most of the park and provides the park with some of its most spectacular scenery. The rock of the escarpment is old. 400 million years ago, this area was covered by a shallow tropical sea teeming with life in the form of plant-like animals, living corals and mollusks. It would have looked much like the present-day Great Barrier Reef of Australia; when the sea began to dry up, the minerals dissolved in it became more concentrated. Magnesium in the water was absorbed into the limestone, which became a harder different sort of rock, called dolomite; the harder dolomite forms much of the rock of the escarpment cliffs along Bruce Peninsula National Park's Georgian Bay shoreline. At Niagara Falls, the dolomite "caprock" is more resistant to erosion than the rock below it, creating the sculptured cliffs for which the area is famous.

Since the last Ice Age, water levels in the region have undergone great changes. Softer limestone has been eroded away by water action, leaving magnificent overhanging cliffs at various points along the shore; these are the big attraction of the Cyprus Lake trails. Where erosion has cut more caves have been formed, such as the famed "Grotto" on the shore between the Marr Lake and Georgian Bay Trails. Great blocks of dolomite, undercut by wave action, have tumbled from the cliffs above and can be seen below the surface of the deep, clean waters of Georgian Bay; the park has a maritime climate with mild winters. In the northern parts of the Peninsula, the climate is among the most temperate in Canada; the climate of park is influenced by both Georgian Bay and Lake Huron, which moderate temperatures. As a result, they tend to prolong milder temperatures in cooler temperatures in spring. Summers are warm, with an average temperature of 16.8 °C while winters are cool, averaging −6.7 °C. Summers are dominated by humid air masses from the Pacific Ocean and the Gulf of Mexico.

In winter, Pacific air masses predominate, bringing in warm and humid air although cold, dry air from the Arctic highs can occur, bringing in colder and drier conditions. Warm air masses coming from the Gulf of Mexico are rare during winter but are responsible for bringing January and February thaws. Spring and fall are characterized by complex weather patterns with contrasting and changing influences from the different regional air masses; the park receives 900 mm of precipitation per year. This is evenly distributed throughout the year with fall being the wettest. Precipitation is lower than inland areas due to the limited influence that the narrow peninsula has when air masses travel over it compared to more interior locations. Animals that inhabit this national park are chipmunks, red foxes, coyotes, black bears, snowshoe hares, white-tailed deer and frogs. In 2006, a new visitors' centre opened to serve Fathom Five National Marine Park and the Bruce Peninsula National Park. Designed by Andrew Frontini of Shore Tilbe Irwin + Partners, the CAD $7.82 million centre, approached by a boardwalk, features an information centre, reception area, exhibit hall and theatre.

A 20-metre viewing tower was constructed to provide visitors with aerial views of the surrounding park and Georgian Bay. The centre was designed with environmental sustainability in mind, receiving $224,000 from the Federal House in Order initiative for implementation of innovative greenhouse gas reduction technology. National Parks of Canada List of National Parks of Canada Official Site Photo gallery and travel information

The Assignment (1997 film)

The Assignment is a 1997 spy action thriller film directed by Christian Duguay and starring Aidan Quinn, with Donald Sutherland and Ben Kingsley. The film, written by Dan Gordon and Sabi H. Shabtai, is set in the late 1980s and deals with a CIA plan to use Quinn's character to masquerade as the Venezuelan terrorist Carlos the Jackal; the film opens to the sounds of a couple having sex. Afterwards, Carlos the Jackal kills a spider in its web with his cigarette and evicts the woman from his room because he claims he has work to do, he is seen donning a disguise, he walks to a cafe where CIA officer Jack Shaw is sitting at a table outdoors. He asks for a light. Shaw does not recognize Carlos, because of his disguise, but he turns to watch Carlos enter the cafe, he watches as Carlos detonates killing dozens of people. The film shows an attack on the OPEC meeting by the Jackal and his followers in 1975. Shaw is sent by the CIA to identify Carlos, but secretly plans to assassinate him with a concealed pistol.

The plan is foiled when his CIA superior stops Shaw from reaching out to shake Carlos' hand because he might be photographed doing so by nearby journalists. In 1986 a man, looking like Carlos, is apprehended in an open-air market in Jerusalem and brutally interrogated by a Mossad commander named Amos; the man claims to be a US Naval officer named Annibal Ramirez whose identification was lost in the chaos of his arrest. Amos confirms his identity and lets him go, stunned that Ramirez looks like Carlos. Back at home, Ramirez is visited by Shaw who tries to recruit him to impersonate the terrorist leader. Ramirez, however, is embittered by his rough treatment at Amos' hands, threatens to sue. Shaw persists, turning up at a navy ball and trying various manipulations to goad Ramirez into taking the assignment, he succeeds by confronting Ramirez with the human cost of Carlos' terrorism, taking him to Bethesda Naval Hospital where a boy has been crippled by one of Carlos' bombs. Amos and Shaw train Ramirez at a former prison in Canada.

Much of his training is devoted to situational internalizing details of Carlos' life. His training concludes with Carla, one of Carlos' ex-mistresses, training Ramirez in how to make love like Carlos; the plan revolves around convincing the KGB, financing his terrorism, that Carlos has begun selling information to the CIA's Counter-terrorism Division. Shaw lures one of Carlos' girlfriends to Libya, where Ramirez meets up with her posing as Carlos during their lovemaking; the girlfriend has become an informant for French intelligence, however. Several French agents arrive at their apartment, Ramirez is forced to kill them in self-defense, he is horrified at having to kill allies in his undercover operation. Carlos sends an assassin to kill the girlfriend in France, ordering him to leave Europe through London; the assassin happens to be in Heathrow airport at the same time as Ramirez, he realizes he is an impostor after Ramirez fails to give the correct response to a password. The assassin forces Ramirez into a struggle ensues.

Amos manages to kill the assassin before being fatally shot. After Amos' death, the CIA suspends the operation pending a review, Ramirez returns home. Back with his wife, he makes love to her as Carlos would, she is disturbed by the change in his personality; the next day, at his son's little league game, he gets into a confrontation with another father and nearly kills him. Shaw bails him out of jail, both men are suffering by not being able to finish their mission and kill Carlos. Ramirez accuses Shaw of fabricating the scene at Bethesda Naval Hospital. Shaw threatens to use the Ramirez family as bait to lure out Carlos; that night Ramirez confesses everything to his wife, including his infidelity, but leaves to go on the mission, knowing that his family will never be safe as long as Carlos is alive. They head to East Berlin and conspicuously meet with each other; the KGB assumes Carlos has been turned. Enraged, the KGB raids Carlos' safe house, but as they try to kill the terrorist, he fatally shoots several agents and escapes.

Shaw and Ramirez are waiting outside for him, Ramirez fights Carlos on the bank of the Spree River. It's impossible to tell; as one of the men is being held under water by the other, Shaw comes upon them and shoots the man above the water several times. He realizes too late that he has shot Ramirez, Carlos swims away. Ramirez presses Shaw to leave him and kill Carlos, but Shaw insists that their plan has worked and that Carlos is now a marked man by the KGB. One way or another, Shaw points out. After returning home, the Ramirez family is seen leaving for Mass, their car explodes in a fireball, Carlos is shown receiving in the mail a newspaper clipping of their deaths. After their funeral we find the deaths of Ramirez and his family were staged by Shaw, in the final scene, the family are safely cavorting on a beach in the Caribbean. Ramirez moves to kill a spider in its web with his cigarette, but changes his mind, he watches his children at peace. An epilogue reveals that Carlos the Jackal was jailed in 1994, after being refused sanctuary by several countries.

A former U. S. Naval officer and a retired CIA agent are rumored to have played a major role in his capture; the Jackal The Assignment on IMDb The Assignment at Rotten Tomatoes The Assignment at AllMovie The Assignment at Box Office Mojo