Rep. Frank R. Wolf (R-Va.) began his bid for a fourth term representing Northern Virginia's 10th Congressional District last night, a $1 million race that is widely expected to be his toughest and most expensive campaign.

Pledging to continue his support of local issues, Wolf kicked off his race against Democrat John G. Milliken of Arlington County before cheering supporters at the Dunn Loring fire station in the district's pivotal Fairfax County.

"We have kept our promise to work for responsible change at the national, regional and local level," Wolf told the crowd of about 400. "Hard as we have worked to achieve the progress we have," he said, "we must work even harder to make certain this progress continues."

A 47-year-old lawyer known for his dogged determination to press issues that some have called mundane, Wolf has said he will stress his work regarding National and Dulles International airports, the rebuilding of the Filene Center at Wolf Trap Farm Park, increased pay for federal workers, and more federal funding for Metro.

Milliken, 40, a Washington lawyer and a member of the Arlington County Board, has said he believes that Wolf, who was first elected in the 1980 Ronald Reagan landslide, may be vulnerable this fall. "I expect it to be a close race, a hard-fought contest, but not an unfriendly one," Milliken said yesterday.

"I'm happy to have him run on his record, because that's the issue," the Democrat said. "The issues are his effectiveness on behalf of Northern Virginia and whether he's in step with the mainstream thinking of Northern Virginia. I think the answer is no."

Republicans tend to agree that the race likely will be the closest congressional contest in Virginia this fall. Wolf campaign Chairman Judith H. Shreve has said that it "is going to be probably the toughest race we've looked at since Frank's election in 1980."

The congressman is expected to spend $1 million, double the amount that Milliken, a law partner of Walter Mondale, has said he will spend. Wolf campaign manager Charles White said the Republicans will have raised nearly $400,000 by the end of the month.

With 55 percent of the district's voters, Fairfax County is expected to be heavily courted by both men. Fairfax has given Wolf, a resident of the Vienna area, increasingly wider margins since he narrowly won election in 1980.

Voters in the northern half of the county, which is in Wolf's district, have proven to be independent, swinging to the Democratic Party's state ticket last year.

Supervisor Nancy Falck, the Republican who represents Fairfax's northernmost Dranesville district, said she expects the county to be "very solidly" for the GOP this year. "I think Mr. Milliken is going to be very hard pressed to say he could do as well or better than Frank Wolf," she said.

Wolf, who ran for the 10th District seat for five years before he unseated Democrat Joseph L. Fisher, is regarded as one of the area's most determined campaigners. "I've wanted to be a congressman since the third grade in Patterson Elementary School, back in Philadelphia," he said recently as he walked across the Capitol grounds.

"This is the pinnacle," he said. "I'm not interested in being senator or governor. I really love this job, and it's fulfilling . . . . I can do things here. I can watch '60 Minutes' and come in the next day and do something about it. You can impact in such a way."

To reach Congress, Wolf said, he had to conquer stuttering. "What I used to hear a lot when I was little, when I was in high school and even in college, was: 'How are you going to run for Congress if you can't even stand up and talk in class?' "

He entered several speech therapy programs and a Dale Carnegie course. "I had to work on it and work on it," Wolf said. "A handicap can give you an incentive to overcome it. You can almost use it as a pole vault to do something, because it gives you greater determination."

Wolf said he is determined to cut into any edge Milliken may have in that rapidly growing county and in Arlington, where the Democrat collected an unprecedented 70 percent of the votes in 1984 in winning another term on the County Board.

In addition to Loudoun and Arlington counties, the district includes the cities of Falls Church and Fairfax as well as northern Fairfax County